Los Angeles hosts an annual “Grilled Cheese Invitational,” but the origins of the grilled cheese sandwich are up for debate. The name “grilled cheese” sandwich is cited in a New York State newspaper in 1928 and in a Texas newspaper in 1930.
 
Bread and cheese is a natural combination, and “ramekins” (a bread and cheese dish) is cited from 1700s.
 
“Toasted cheese sandwiches” were on menus from the 1900s and were popularly cited in newspapers by 1910. The Wikipedia (below) claims that the sandwiches were served “open face” until the 1960s, but 1910s citations show some “closed” sandwiches.
 
“Grilled American Cheese” sandwich is often abbreviated “GAC” and pronounced and written as “Jack.”
   
   
Cooks.com
GRILLED CHEESE SANDWICH  
The most important part of any meal is the time and effort you put in to it. To make a great Grilled Cheese Sandwich you need fresh cheese and fresh bread. Ingredients are bread, cheese, and butter.
 
Take 2 pieces of bread, a slice of cheese and insert the cheese in between the bread, butter the outside of the bread on each side. Place in a pan on low heat and turn when needed, until both sides are golden brown. Serve with pickles and potato chips on the side.
 
Cooks.com 
GRILLED CHEESE SANDWICH  
2 slices bread
2 slices American cheese
Butter
1 napkin
 
Toast bread in toaster. Spread butter on both sides of toast. Place cheese slices between toast. Place on napkin in microwave. Microwave 30 seconds on high.
 
Wikipedia: Grilled cheese sandwich
A grilled cheese sandwich, (also known as cheese toasty or toasted cheese sandwich) is a form of toasted sandwich that consists of two slices of bread and at least one slice of cheese melted in between.
 
There are various methods of preparation which vary depending on taste and convention. In the United States it is most common for the assembled sandwich to be buttered on the outside and placed on a griddle, pan, cast iron skillet, or dedicated sandwich maker to be heated (technically, the word ‘grilled’ in the name of the meal is inaccurate). Once the bread on the bottom half of the sandwich has reached a toast-like texture, the sandwich is flipped, and continues cooking until the other side has toasted or the cheese has melted. Another method of cooking the grilled cheese is to butter one slice of bread on both sides, grill the bottom side until the butter melts, then flip. Wait for the inside to be grilled, flip and add the cheese. Then butter the other slice of bread and grill then place the slices of bread together. In the United Kingdom the sandwich would most commonly be grilled either (open face as cheese on toast) under a grill (broiler) or, more usually, in a toasted sandwich maker. In the UK the sandwich would not normally be cooked in the American fashion, using a frying pan. Cheddar is most commonly used whilst in the US, traditionally American cheese is used; however, other cheeses such as gouda, Velveeta cheese and Swiss can be used. It is often supplemented with additional ingredients, most notably bacon and/or tomato, but also avocado, tuna, meat, pickles, herbs, spices, and condiments such as mustard, tomato sauce, or Worcestershire sauce. It is traditionally accompanied by tomato soup. Some recipes call for grilled cheese sandwiches to be served with ketchup as the condiment of choice.
 
The grilled cheese sandwich is a variation on the very old combination of bread and cheese. The modern grilled cheese sandwich (American cheese and sliced white bread) began in the 1920s as an open sandwich. The additional slice of bread became common in the 1960s. In the early 1970s, a Canadian restaurant owner by the name Dach Johnston was credit for the addition of bacon to the sandwich, however, evidence of this has not yet been proven. 
 
(Oxford English Dictionary)
ramekin
A small quantity of cheese, with bread-crumbs, eggs, etc., usually baked and served in a special mould. Chiefly pl.
1706 PHILLIPS, Ramequin (Fr. in Cookery), toasted Cheese and Bread, a Toast and Cheese. Ramequins are also small slices of Bread-crum cover’d with a Farce made of pounded Cheese, Eggs and other Ingredients bak’d in a Pie-pan.
1754 Connoisseur No. 19 Toasted cheese is already buried in rammelkins. 
 
Feeding America
Jennie June’s American Cookery Book
by Jane Cunningham Croly
New York, NY: The American News Co.
1870
Pg. 108:
TOASTED CHEESE.—1.
Grate three ounces of fat cheese, mix it with the yolk of one egg, four ounces of grated bread and three ounces of butter, beat the whole well in a mortar, with a dessert-spoonful of mustard, and a little salt and pepper. (Pg. 109—ed.) egg, four ounces of grated bread and three ounces of butter, beat the whole well in a mortar, with a dessert-spoonful of mustard, and a little salt and pepper. Toast some slices of bread, lay the paste thick upon it, put it for a minute before the fire and send to table very hot.
TOASTED CHEESE.—2.
Put into a clean sauce-pan a table-spoonful of either ale (not bitter) or cold water; add some slices of toasting cheese, and let it simmer until it is melted, stirring it all the time. Have ready in a bowl some good ale, sweeten it to the taste with moist sugar and add some grated nutmeg. Toast slices of bread without either burn or crust, put them hot into the bowl, to take the chill off the ale, then put a slice of the toast on a hot plate for each person, and pour upon it as much of the cooked cheese as may be agreeable. Take out of the bowl any remaining toast there may be left; stir well the sugar from the bottom, and drink the ale after eating the cheese.
 
Google Books
Luncheons: A Cook’s Picture Book
by Mary Arnold
New York, NY: The Century Co.
1902
Pg. 209:
TOASTED CHEESE SANDWICHES
Make a filling of grated cheese, toast the sandwiches on both sides, and serve them hot. 
   
Google Books
Louis’ Every Woman’s Cook Book
by Louis Muckensturm
New York, NY: Dodge Publishing Company
1910
Pg. 104:
TOASTED CHEESE SANDWICHES
Cut and toast six slices of bread. Cover three of them with thin slices of American cheese, and put in the oven in a tin pan. When the cheese is nearly melted, place the other slices of toast on top; keep the sandwiches for two more minutes in the oven to have them very hot. Trim off the crust and serve in a covered dish.
       
14 June 1910, Syracuse (NY) Post-Standard, pg. 12, col. 1:
A quick dainty to whip up in the chafing dish is toasted cheese sandwiches, which are not widely known. Wheaten bread is used for these. Cut in rather thick slices and trim off the crusts.
 
Between each two slices put a thick slice of cheese. Season, if liked, with salt and paprika. Melt in the blazer, a tablespoonful of butter, and when it reaches bubbling point lay in as many sandwiches as the dish will hold at one time conveniently. There must be room to turn them. Have the hot plates ready nearby. When the bread is brown on one side, turn and brown on the other. By this time the cheese will be melted between them and the sandwiches are ready to be enjoyed.
 
29 March 1911, Stevens Point (WI) Gazette, pg. 14, col. 3:
Toasted Cheese Sandwiches.
Make round sandwiches (round loaf is best) with slice of thin rich cheese the same size of the thin sliced bread. Brown butter in chafing dish and fry a good brown. Serve at once on individual plates. Fine for afternoon tea.
 
19 June 1913, Waukesha (WI) Freeman, pg. 3, col. 4 ad:
Have You Tried Our
Toasted Cheese Sandwiches?
 
26 November 1913, Grand Forks (ND) Daily Herald, pg. 8:
Toasted Cheese Sandwiches.
Plain bread and butter with slices of cheese put between as sandwiches are toasted or browned in the chafing dish after frying bacon.
 
Google Books
Meatless Cookery
by Maria McIlvaine Gillmore
1914
Pg. 210:
TOASTED CHEESE SANDWICHES
Plain bread and butter sandwiches with fairly thick slices of cheese put between the slices are toasted, or at chafing-dish suppers may be browned in a pan. Use a slow heat that the cheese may be melted when the bread is browned. 
   
4 August 1914, Wilkes-Barre (PA) Times-Leader, pg. 10:
Toasted Cheese Sandwiches
Cut slices of bread a little more thickly than is usual for sandwiches, spread 1 slice with butter and the other with a filling made by grating or grinding through a food chopper 1/2 pound of cheese, to which may be added any desired seasonings, as, for instance, a few drops of onion juice, a little butter if the cheese is dry, a little fresh or canned pepper or a sprinkling of nuts. Spread the filling generously on the bread, press the two slices firmly together and toast the outer surfaces, during which process the cheese will slightly melt. Serve very hot, not preparing the sandwiches until ready to place them on the table.
   
29 August 1917, Philadelphia (PA) Inquirer, pg. 7:
TOASTED CHEESE SANDWICHES
Prepare toast, dip in hot salted water. Spread slice with grated cheese; place in a pan in oven long enough to melt the cheese. Put slices together as sandwiches.
 
Feeding America
The International Jewish Cook Book
by Florence Kreisler Greenbaum
New York, NY: Bloch Publishing Company\
1919
Pg. 9:
TOASTED CHEESE SANDWICHES
The filling for the toasted cheese sandwiches calls for a cup of soft, mild cheese, finely cut, and stirred over the fire with a tablespoon of butter until the cheese is melted. Enough milk to moisten, perhaps not more than one-eighth of a cup, is then added, with salt, mustard and paprika to taste, and the whole is stirred until creamy and smooth. Slices of bread are very thinly buttered, the cheese mixture spread on generously, each slice covered with another slice, and set away until the filling cools and hardens, when the sandwiches are toasted on both sides and served hot.
 
Google Books 
Castelar Crèche Cook Book
by Board of Directors, Castelar Crèche Home for Homeless Babies
Los Angeles, CA: Times- Mirror Printing and Binding House
1922
Pg. 255:
TOASTED CHEESE SANDWICH
Remove crusts from slices of bread and then butter the slices thinly. Spread 1 slice with finely minced ham and place over this a slice of American cheese sprinkled with cayenne pepper and salt, then add another slice of bread, more cheese, properly seasoned, and finally a third slice of bread. Toast these under a gas broiler and when the cheese is melted, cut diagonally, making triangular sandwiches. 
 
22 April 1929, Syracuse (NY) Herald, section X, pg. 16, col. 7:
Dishes That May Be Prepared on an Electric Grill.
1. All types of hot sandwiches such as grilled cheese, tongue and egg, club, sardine, etc. 
 
Google Books
The Edgewater Sandwich Book
by Arnold Shircliffe
Chicago, IL: J. Willy, Inc.
1930
Page 156
Toasted Cheese Sandwich.

7 June 1930, Galveston (TX) Daily News, pg. 14, col. 1:
AUTO LUNCHES FOR
THE WEEK-END TRIP
   
GRILLED CHEESE SAND-
WICHES DELIGHTFUL WHEN
MADE IN THE OPEN
(...)
Grilled cheese sandwiches are delightful when made in the open. Lay large, thin slices of cheese between two slices of buttered bread. Either toast these on the end of a green stick or a gridiron or saute them in bacon fat in a saucepan. Those who take long hikes will be interested to know that if cheese is wrapped in a cloth moistened with vinegar, it will not mold so quickly.
 
25 January 1934, Milford (Iowa) Mail, pg. 2, col. 5:
Miriam Lowenberg, of the foods and nutrition department at Iowa state has a special grilled cheese sandwich that goes straight to the heart. She spreads whole cream cheese on unbuttered bread, adds a bit of prepared mustard to which a drop of Worcestershire sauce has been added, then—and this is what makes it different—she butters the outside of the sandwich lightly and toasts it in the broiler oven, or browns it quickly in a heavy skillet.
   
21 February 1934, Huntingdon (PA) Daily News, pg. 7, col. 5:
Grilled cheese sandwiches are nice, but you’d be surprised at the amount of difference added to the taste if halved walnuts are pressed into the cheese while the sandwich is warm. It’s decidedly an improvement.

31 January 1935, The Garfieldian (Chicago, IL), pg. 6 ad:
MALTED MILK and
Grilled CHEESE SANDWICH
25c
(Walgreen Drug Store—ed.)
   
29 May 1935, Huntingdon (PA) Daily News, pg. 7, col. 6:
Grilled Cheese Sandwiches
 
4 October 1935, North Adams (MA) Transcript, pg. 9, col. 6 ad:
Grilled Cheese Sandwich and Hot Chocolate, 20c
   
6 March 1936, Hammond (IN) Times, pg. 19, col. 5:
Grilled Cheese Sandwiches
1 hard-cooked egg
1 1/2 tablespoons pimento
1 small onion
1/2 pound American cheese
1/2 cup evaporated milk
1/2 teaspoon salt
1 tablespoon butter
1 tablespoon flour
1 tablespoon Worcestershire sauce
Put the egg, pimento and onion with the cheese through a food chopper using the medium knife. Melt the butter in a saucepan and add the flour with the seasoning. Add the milk and stir until the mixture comes to a boil. Combine this mixture with the cheese mixture and spread between slices of bread. Butter the outside of the bread and toast under a low flame until golden brown.
 
10 February 1938, Wisconsin Rapids (WI) Daily Tribune, pg. 7, col. 5 ad:
GRILLED CHEESE SANDWICH
A tasty, filling delicious light lunch 14c
(Ford Hopkins Drug Store—ed.)
 
Live Search Books
New York Panorama
by Federal Writers’ Project, New York City
New York, NY: Random House
1938
Pg. 159 (The Local Vernacular):
...smear one, burn it for a toasted cheese sandwich;...
 
New York (NY) Times
No Glamour, but Sandwich Is a Star
By JENNIFER STEINHAUER
Published: October 3, 2007
(...)
Thursday is grilled cheese night at Campanile, a standard-bearer of Italian dining in Los Angeles, and the restaurant’s busiest night, when the tables bustle with families, hot daters, girls-night-out revelers downing prosecco, and divorced dads hoping to buy good will from their estranged children.
 
The Melt Down, a restaurant in Culver City devoted to the gooey sandwich, has lines out the door at lunch. Every April is grilled cheese month at Clementine, a lunch spot near Century City, with an elaborate new theme each time.
 
For the past four years, this city has also been home to the Grilled Cheese Invitational. Roughly 600 people show up at an unpublicized address, armed with frying pans and camping stoves, and are given 20 minutes to demonstrate their grilled cheese prowess. (One year, a contestant constructed an eight-foot grilled sandwich rendition of “The Gates” by Christo and Jeanne-Claude.)