Captain Hanson Crockett Gregory of Maine (1831-1921) is usually given credit for inventing the doughnut hole while at sea in 1847. Gregory explained his invention in an interview given in 1916 (see below).
 
Doughnuts with holes are cited in print from at least 1861.
 
 
Wikipedia: Doughnut
A doughnut (also spelled donut), is a sweet, deep-fried piece of dough or batter. The two most common types are the torus-shaped ring doughnut and the filled doughnut, a flattened sphere injected with jam, jelly, cream, custard, or other sweet filling. A small spherical piece of dough, originally made from the middle of a ring doughnut, may be cooked as a doughnut hole. Baked doughnuts are a variation that is baked in an oven instead of being deep fried.
 
Overview
Ring doughnuts are formed either by joining the ends of a long, skinny piece of dough into a ring or by using a doughnut cutter, which simultaneously cuts the outside and inside shape, leaving a doughnut-shaped piece of dough and a doughnut hole from dough removed from the center. This smaller piece of dough can be cooked or re-added to the batch to make more doughnuts. A disk-shaped doughnut can also be stretched and pinched into a torus until the center breaks to form a hole. Alternatively, a doughnut depositor can be used to place a circle of liquid dough directly into the fryer. Doughnuts can be made from a yeast-based dough for raised doughnuts or a special type of cake batter. Yeast-raised doughnuts contain about 25% oil by weight, whereas cake doughnuts’ oil content is around 20%, but they have extra fat included in the batter before frying. Cake doughnuts are fried for about 90 seconds at approximately 190°C to 198°C, turning once. Yeast-raised doughnuts absorb more oil because they take longer to fry, about 150 seconds, at 182°C to 190°C. Cake doughnuts typically weigh between 24 g and 28 g, whereas yeast-raised doughnuts average 38g and are generally larger when finished.
 
After being fried, ring doughnuts are often topped with a glaze (icing) or a powder such as cinnamon or sugar. Styles such as fritters and jelly doughnuts may be glazed and/or injected with jam or custard.
 
As well as being fried, doughnuts can be completely baked in an oven. These have a slightly different texture from the fried variety with a somewhat different taste due to the lack of absorbed oil—and so have a lower fat content.
 
There are many other specialized doughnut shapes such as old-fashioneds, bars or Long Johns (a rectangular shape), or with the dough twisted around itself before cooking. In the northeast USA, bars and twists are usually referred to as crullers. Doughnut holes are small spheres that are made from the dough taken from the center of ring doughnuts or made to look as if they are. These holes are also known by brand names, such as Dunkin Donuts’ Munchkins and Tim Hortons’ Timbits.
 
History
Possible origins
Doughnuts have a disputed history. One theory suggests that doughnuts were introduced into North America by Dutch settlers, who were responsible for popularizing other American desserts, including cookies, cream pie, and cobbler. This theory is bolstered by the fact that in the mid-19th Century doughnuts were called by the Dutch olykoeks (“oily cakes”). However, there is also archaeological evidence that the pastries were prepared by prehistoric Native Americans in southwestern USA.

Hansen Gregory, an American, claimed to have invented the ring-shaped doughnut in 1847 aboard a lime-trading ship when he was only sixteen years old. Gregory was dissatisfied with the greasiness of doughnuts twisted into various shapes and with the raw center of regular doughnuts. He claimed to have punched a hole in the center of dough with the ship’s tin pepper box and later taught the technique to his mother.
 
Wikipedia: Donut Hole (Medicare)
The term Donut Hole (or Doughnut Hole) refers to a “coverage gap” within the defined standard benefit under the Medicare Part D prescription drug program. Under the defined standard benefit package there is a gap in coverage between the initial coverage limit and the catastrophic coverage threshold. Within this gap, the beneficiary pays 100% of the cost of prescription drugs before catastrophic coverage kicks in. The term “coverage gap” is preferred by Centers for Medicare and Medicaid Services (CMS) and Prescription Drug Plans, but Donut Hole has been more widely adopted in the popular media.
 
Google Books
Merry’s Museum,
Parley’s Magazine, Woodworth’s Cabinet, and The Schoolfellow
Edited by Robert Merry and Hiram Hatchet
Volumes XLI and XLII
New York, NY: J. N. Stearns
1861
Pp. 44-45:
Her brother James, who never talked anything but nonsense when he could help it, declared she ate nothing but the hole of a doughnut.
 
Google Books
The Rebellion Record
by Frank Moore
1862
Pg. 73:
THE FEAST OF DOUGHNUTS.—The ladies of Augusta, Me., some time ago distributed over fifty bushels of doughnuts to the Third Volunteer regiment of Maine. A procession of ladies, headed by music, passed between double lines of troops, who presented arms, and were afterwards drawn up in hollow square to receive the welcome doughnation.
 
Neve before was seen such an aggregate of doughnuts since the world began. The circumambient air was redolent of doughnuts. Every breeze sighed doughnuts—everybody talked of doughnuts. The display of doughnuts beggared description. There was the molasses doughnut and th sugar doughnut—the long doughnut and the short doughnut—the round doughnut and the square doughnut—the rectangular doughnut and the triangular doughnut—the single twisted doughnut and the double twisted doughnut—the “light riz” doughnut and the hard-kneaded doughnut—the straight solid doughnut and the circular doughnut, with a hole in the centre. There were doughnuts of all imaginary kinds, qualities, shapes, and dimensions. It was emphatically a feast of doughnuts, if not a flow of soul.—Baltimore American, June 19. (1861—ed.)
       
Google Books
Summer Rest
By Gail Hamilton
Boston, MA: Ticknor and Fields
1866
Pg. 333:
They just drew a circle on the greensward, and cut out a deep round hole as clean and regular as the hole in a doughnut before it is cooked ; no jagging into the turf, no scattering about of stones and soil, but a round hole constantly deepening, a pyramidal mound constantly rising.
     
18 March 1886, Boston (MA) Daily Globe, pg. 4:
Doughnuts Whole and Doughnut Hole.
 
Google Books
Miss Parloa’s Kitchen Companion
by Maria Parloa
Nineteenth Edition
New York, NY: The Clover Publishing Co.
1887
Pg. 50 (illustration):
Doughnut-cutter.
 
Google Books
7 January 1888, Good Housekeeping, “Clinton Doughnuts,” pg. 113:
Roll this down to the thickness of half an inch and cut into round cakes with a hole in the center. If you do not possess a regular doughnut-cutter, a biscuit-cutter will do, as a piece can be cut from the center with a thimble.
     
8 March 1888, Cambridge City (IN) Tribune, pg. 4, col. 3: 
The health journals and the doctors all agree that the best and most wholesome part of the ordinary New England country doughnut is the hole. The larger the hole, they say, the better the doughnut.—Journal of Education.
 
Google Books
March 1889, The Peoria Medical Monthly, pg. 382:
I have used the ordinary syringe stem with a rubber shield shaped like a doughnut, the central hole being quite small. 
 
Google Books
15 February 1897, Gleanings in Bee Culture, pg. 114:
Roll about half an inch thick, and cut out with a doughnut or jumble cutter, which leaves a hole in the center.
 
Google Books
9 October 1897, Notes and Queries, pp. 294-295:
CAKES (8th S. xii. 8,58,98)—The cakes spoken of by MR. JEAKES are the doughnut, familiar in American stories. The ingredients and proportions of doughnuts are those of the Channel Island “wonders” (or by the peasants called merveilles). When I ate them there I found no difference except in shape. The American doughnut is round, with a round hole in the middle—a thick ring, in fact. Fifty years ago, in Maine, U.S., I saw them shaped very much like those of the Channel Islands. They are certainly of continental origin, for the Dutch women of New Amsterdam (New York) excelled in making them. Out of New England they are usually called fried cakes.
W.
Boston, U.S.
 
22 April 1913, Kansas City (MO) Star, pg. 11:
WHY IS A DOUGHNUT HOLE?
A MOST ESSENTIAL PART OF THE
DAINTIES, EXPERTS SAY.
Without the Aperture, Says Adriene Del-
vaux, Chef at the Hotel Baltimore,
The Result Is a Soggy Piece
of Cooked Dough.
Why is the hole in the doughnut?
 
At first glance it looks like a foolish question, But it isn’t. It’s a serious question that does nothing but bring respect for the mechanical ability of cooks. There have been several answers to the “Public Mind” question, all wrong.
 
There’s a reason for the hole. In the first place, if there weren’t a hole in the doughnut it wouldn’t be a doughnut at all. it would be a “Bismark.” That’s what Adriene Delvaux, chef at the Hotel Baltimore, says.
 
BIGGER THE HOLE THE BETTER.
Here’s the proposition: The biger the hole the better the doughnut. Doughnuts are cooked in hot grease. If the hole is large there is a greater area of the doughnut’s surface directly exposed to the grease. Therefore the cooking is done quickly and thoroughly. The desired crispiness results. It’s perfectly plain.
 
Mrs. E. J. Ferguson, manager of The Star’s cafe, knows a lot about the doughnuts. Mrs. Ferguson made some doughnuts this morning, more as an experiment than anything else. There were three types, the large holed or European doughnut, the small holed doughnut of America and the German variety without a hole called “Bismarks.”
 
THE PANCAKE DOUGHNUTS UNPOPULAR.
The two with oles cooked in three minutes, but it took an extra half minute for the “Bismark.” The one with the large hole was delicious. It was crisp and brown and a perfect speciment. The one with the small hole looked good but it was soggy. The “Bismark” was awful. The same dough was used with all.
 
“We make doughnuts with holes simply because we can cook them quicker and better,” M. Delvaux said this morning. “It’s a mechanical reason. The dough may be rolled out in the shape of a stick of candy and the ends left unattached, and an exquisite doughnut will result. Once in a while we make a doughnut shaped like a pancake, but they ar not popular at all.”
   
26 March 1916, Washington (DC) Post, pg. ES9:
“Old Salt” Doughnut Hole Inventor Tells Just How Discovery Was Made And Stomach of Earths Saved
Special to The Washington Post.
Boston, March 25.—The man who invented the hole in the doughnut has been found. He is Capt. Hanson Gregory, at present an inmate in Sailor’s Snug Harbor, at Quincy, Mass. Doughnut cutters have made fortunes for men; millions eat doughnuts for breakfast and feel satisfied. Doctors do not assail the doughnut. And all of this owes its being to Capt. Gregory, who made the doughnut a safe, sane and hygienic food.
   
It’s a long story, mates; but as the 85-year-old chap relates it, it’s only too short. Outside the fact that Capt. Gregory is a bit hard of hearing, he’s as sound as new timber.
   
He’s a product of Maine; and so Maine can lay claim to the discoverer of the hole in the doughnut, along with the discoverer of new ways to evade the prohibition laws. But Capt. Gregory’s discovery is of real use in the world; millions have risen, and millions more shall rise up, and call him blessed.
   
‘Bout ‘47 Was the Date.
“It was way back—oh, I don’t know just what year—let me see—born in ‘31, shipped when I was 13—well, I guess it was about ‘47, when I was 16, that I was aboard ship and discovered the hole which was later to revolutionize the doughnut industry.
   
“I first shipped aboard the Isaac Achorn, three-masted schooner, Capt. Rhodes, in the lime trade. 
Later I joined other crews and other captains, and it was on one of these cruises that I was mawing doughnuts.
   
“Now in them days we used to cut the doughnuts into diamond shapes, and also into long strips, bent in half, and then twisted.  I don’t think we called them doughnuts then—they was just ‘fried cakes’ and ‘twisters.’
   
“Well, sir, they used to fry all right around the edges, but when you had the edges done the insides was all raw dough.  And the twisters used to sop up all the grease just where they bent, and they were tough on the digestion.”
 
“Pretty d—d tough, too!” profanely agreed one of the dozen pipe-smoking fellows who were all eyes and ears, taking in their comrade’s interview by The Post reporter.
   
With a glance at the perfervid interrupter, the discoverer continued:
   
“Well, I says to myself, ‘Why wouldn’t a space inside solve the difficulty?’  I thought at first I’d take one of the strips (Col. 2—ed.) and roll it around, then I got an inspiration, a great inspiration.
   
“I took the cover off the ship’s tin pepper box, and—I cut into the middle of that doughnut the first hole ever seen by mortal eyes!”
   
“Were you pleased?”
   
“Was Columbus pleased?  Well, sir, them doughnuts was the finest I ever tasted.  No more indigestion—no more greasy sinkers—but just well-done, fried-through doughnuts.
   
“That cruise over, I went home to my old mother and father in Camden, Me., where I was born. My father, Hanson Gregory, sr., lived to be 93, and my mother lived to be 79. She was a pretty old lady then. I saw her making doughnuts in the kitchen—I can see her now, and as fine a woman as ever-lived, was my mother.
 
Taught Trick to Mother.
“I says to her: ‘Let me make some doughnuts for you.’ She says all right, so I made her one or two and then showed her how.
 
“She then made several panfuls and sent them down to Rockland, just outside Camden. Everybody was delighted and they never made doughnuts any other way except the way I showed my mother.
   
“Well, I never took out a patent on it; I don’t suppose any one can patent anything he discovers; I don’t suppose Peary could patent the north pole or Columbus patent America. But I thought I’d get out a doughnut cutter—but somebody got in ahead of me.
     
Hole “Cut Out,” His Joke.
“Of course a hole ain’t so much; but it’s the best part of the doughnut—you’d think so if you had ever tasted the doughnuts we used to eat in ‘31. Of course, lots of people joke about the hole in the doughnut.  I’ve got a joke myself:  Whenever anybody says to me: ‘Where’s the hole in the doughnut?’  I always answer: ‘It’s been cut out!’” and the old chap laughed loud and longat his little sally, while the rest joined in.
   
So there he sits—in the Snug Harbor by the sea. And whenever there’s doughnuts on the day’s fare, Capt. Gregory takes a personal pride trying to do what nobody’s succeeded in doing yet—in trying to find the hole in the doughnut. And whenever the old salts rally him about it, he always springs his little joke:
   
“The hole’s been cut out, I guess!” to the delight of the whole shipful.
   
New York (NY) Times 
On Language
Doughnut Hole
By WILLIAM SAFIRE
Published: February 25, 2007
On March 18, 1886, an anonymous reader of The Boston Globe wrote to the editor, “Can a man get fat on a diet of doughnut holes?” The reader promptly answered his own question: “Doughnut holes can only be introduced into the stomach by swallowing the doughnut whole.”
 
Yuk, yuk, went the 19th-century editor, who then courageously printed the letter. This bit of history was provided to On Language by Fred Shapiro, bulldog editor of the Yale Book of Quotations, who set a javert of search engines whirring in responding to my query for the first printed use of doughnut hole.
 
The etymology of this phrase is important because its current use has become a source of worry for millions of advancing years. (That’s the gentlest euphemism I can find for “incipient old folks”; though “advancing years” has an ominous overtone.)
 
That’s because Representative Nancy Pelosi of California, one month before the recent election (“the late unpleasantness,” as Republicans like to put it), promised voters at a senior center in Florida, “We will use that money to fill the doughnut hole so that seniors will have affordability, they will have reliability and will not be caught in this trap of the doughnut hole.” The money that the future speaker of the House was talking about would come, she argued, from savings brought about by future government drug-price negotiations and would be used to close the gap in Medicare coverage known far and wide as the doughnut hole.
 
Whence this locution? It is the figurative gap in coverage that requires many people in Medicare Part D to pay for drugs themselves. For this year, the new benefit has Medicare covering 75 percent of the first $2,400 a person spends on drugs (after a $265 deductible), but after that is spent, coverage pauses until the yearly expenses reach $5,451, at which point Medicare coverage kicks back in. Why the gap? The Washington Post describes proponents seeing it “as a way to provide some help to all beneficiaries and substantial help to those with catastrophic drug costs and yet not break the bank with the new benefit.”