Entry in progress—B.P.
 
Wikipedia: Pot roast (beef)
Pot roast is a braised beef dish. Pot roast is typically made by browning a roast-sized piece of beef (often taken from the tougher chuck cut) to induce a Maillard reaction, then slow-cooking in an acidulated liquid in a covered dish.

Boneless chuck roast and 7-bone pot roast are recommended,[by whom?] as they are inexpensive, yet become tender after braising. As with all braises, the slow cooking tenderizes the tough meat, while the liquid exchanges its flavor with that of the beef. The result of a good pot roast should be tender, succulent meat and a rich liquid that lends itself to gravy.

Pot roast is often served with carrots and/or potatoes simmered in the cooking liquid.
 
Merriam-Webster Dictionary
Main Entry: pot roast
Function: noun
Date: 1881
: a piece of beef cooked by braising usually on top of the stove
 
(Oxford English Dictionary)
pot roast, n.
orig. and chiefly U.S.
[< POT n.1 + ROAST n.]
A dish consisting of a piece of meat, typically beef, cooked slowly in a covered dish.
1880 N.Y. Times 11 Apr. 9/2 Sour Braten, or a Sour Pot-roast… Take a nice piece of beef from the round, rub well with salt and pepper and lay it in a vessel, [etc.].
1881 F. E. OWENS Cook Bk. 59 Pot Roast of Beef. Get a solid piece from the round, [etc.].
1897 Altrurian Cook Bk. 38 (heading) Mutton Pot Roast.
1929 E. WILSON I thought of Daisy iv. 232 The dinner was an admirable pot-roast, with onions, potatoes, and carrots.
1936 L. C. DOUGLAS White Banners iii. 59 ‘We’ve had breast of lamb three times in the past two weeks.’.. ‘Sorry,’ said Hannah. ‘I’ll have a pot roast to-morrow.’
1993 Amer. Way 15 Nov. 145/1 They have taught us that Yankee pot roast and fried chicken can be taken just as seriously as French peasant classics such as cassoulet and choucroute.
 
7 February 1877, Springfield (MA) Republican, pg. 3:
POT ROAST.—Meat of any kind, beef, chickens, prairie fowl or pigeons, may be cooked in this way; Slice an onion and a few slices of pork, and put in the bottom of a kettle. Place on top whatever meat is to be cooked, and just water enough to stew it, no more. Keep turning the meat, and lket it stew or rroast slowly to brown and tender; and then take it out, strain and thicken the gravy, pour over the meat, and serve hot.
 
Feeding America
Mrs. Lincoln’s Boston Cook Book:
What To Do and What Not To Do in Cooking
By Mary Johnson Bailey Lincoln
Boston, MA: Roberts Brothers
1884
Pg. 227:
Smothered Beef, or Pot Roast.
Four to six pounds from the ,i>middle or face of the rump, the vein, or the round. Wipe with a clean wet cloth. Sear all over by placing in a hot frying-pan and turning till all the surface is browned. Put in a kettle with one cup of water, and place it where it will kepp just below the boiling-point. Do not let the water boil entirely away, but add only enough to keep the meat from burning.
Pg. 228:
Have the cover fitting closely to keep in the steam. Cook until very tender, but do not let it break. Serve hot or cold. The meat when cold is delicious, cut in quarter-inch slices, and sauted in hot butter.
 
17 October 1885, New Haven (CT) Register, “Receipts from Miss (Maria) Parloa,” pg. 2:
POT ROAST.—A tough piece of meat can be made very savory and tender by this mode of cooking. WIth a piece weighing between six and seven pounds use the following named ingredients: Three heapong teaspoonfuls of salt, half a teaspoonful of pepper, two cloves, three tablespoonfuls of flour and one quart of water.
 
Wipe the meat with a damp cloth and put it into an iron pot. Sprinkle with salt and pepper. Put over the fire and brown slowly, turning frequently. It will take about half an hour for this cooking. When the meat has been browned sufficiently, put half a pint of boiling water into the pot and cover closely. Set the pot where its contents will simmer for four hours. As the water steams away, add a little more boiling water—about half a pint at a time. At the end of the four hours mix the flour with half a pint of cold water. Take up the meat and skim the fat from the gravy in the pot. Turn the flour and water into the gravy, and, placing the pot in a hot place, stir the gravy until it boils. Cook for five minutes. After satiisfying yourself that the gravy is seasoned to your taste, pour a small quantity over the meat and serve the remainder in a dish. Either boiled rice or potatoes should accompany the roast, beside any other vegetables you choose.
 
10 January 1886, Macon (GA) Weekly Telegraph, pg. 4:
CALLED “POT ROAST.”
Manner of Cooking an Exceptionally Cheap and Tender Dish.
New York World.
Even on a limited income beef should be the principal meat, for it is the most wholesome, other kinds make an agreeable change. Use pork sparingly, though it is cheap. Beef can be made equally cheap by choosing suitable pieces and cooking them well. Take a thick slice of meat from the round, such as you can buy at the market for ten cents per pound, with no bone and little fat. Get the butcher to split it almost open, so that you have one large thin steak. No matter how tough, it will be tender as porterhouse steak when ready for the table and quite as toothsome. Lay the meat out smoothly and wipe it dry, but do not wet it. Take a coffee cup full of fine bread crumbs, a little salt and pepper, a little powdered thyme or other sweet herb, and just enough milk to moisten to a stiff dressing. Mix well and spread over the meat. Roll it up carefully and tie it up with twine, wound to secure it well, especially the ends. Now, in the bottom of your kettle fry some fat salt pork till crisp and brown, one quarter pound cut in thin slices (cost three cents). Into the fat that has filed out from the pork put the rolled meat, brown it on all sides, turning it till it is a rich color all over, then put in half a pint of water and prinkle over a little salt. Keep closely covered, adding a little water if it cooks away too much. If one likes the flavor of onion add the half of a small one, chopped fine. When ready to serve entie the string carefully to preserve the shape. lay it on a platter with the gravy poured over it. Cut the meat in slices through the roll as jelly roll is cut by the bakers. The toughest meat is made tender and nutritious cooked in this way, and is equally nice warmed over the next day.
   
19 May 1901, Philadelphia (PA) Inquirer, first section, pg. 5 ad:
Yankee Pot Roast, Potatoes, Bread and Butter…15c
(Shoneman’s Cafe—ed.)
   
19 October 1915, Kingston (NY)

, pg. 11, col. 5:
MONDAY—Yankee Pot Roast, Julienne, Mashed Potatoes, Green Preas… $.40
(The Weiner Hotel Restaurant—ed.)
 
19 September 1932, New York (NY) Times, pg. 8 ad:
Yankee Pot Roast
With Noodles
Choice of Vegetables
30 cents
at Windsor Cafeterias