A “melt” sandwich is a sandwich that contains melted cheese over its main ingredient. Popular “melts” include the “patty melt” (cited in print from at least 1956 and possibly from California) and the “tuna melt” (cited in print from at least 1966). The “patty melt”—a combination of a hamburger and a grilled cheese sandwich—is sometimes merely another name for “cheeseburger.”
 
In October 2008, the blog A Hamburger Today described the “Fatty Melt”—a hamburger with two grilled cheese sandwiches for buns.
   
 
Wikipedia: Melt sandwich
A melt sandwich is a type of sandwich consisting of bread, some sort of filling, and a layer of cheese, sometimes grated. The sandwich is then grilled or fried until the cheese is melted. It may be served as an open face sandwich or a closed face one.

One common filling is tuna, with or without mayonnaise; the result would then be a tuna melt sandwich. Other popular choices are ham or a hamburger, the latter of which is known as a patty melt. It is the filling that establishes the melt sandwich as a variation of the grilled cheese sandwich.
 
wiseGEEK
What is a Patty Melt?
(...)
It’s unclear when the actual patty melt was invented, but records exist of them being served as early as the 1940s. They embellish on the traditional cheeseburger, and further take the burger back to its roots by being served on bread versus a bun. The patty melt is usually served between two slices of grilled rye bread, just as early hamburgers were served between two pieces of bread.

The classic patty melt is a pretty wonderful thing if you’re a fan of hamburgers and onions. Traditionally the sandwich includes one hamburger patty, which is topped with swiss cheese. Grilled onions are added and the whole burger with rye bread included is grilled to fully melt the cheese. The result is a hot, flavorful burger that differs from the normal hamburger patty. Substitutions for rye bread include panini, Texas toast, or any type of white bread, and some burgers can be made of turkey.
 
Typically patty melts are served with lettuce, tomato and pickles on the side, and don’t have condiments on the bread. Some restaurants that serve this popular diner fare slather the inside of the bread with thousand island dressing, but this is not usual. You can certainly request condiments on the patty melt, but the flavor of the onions often renders them unnecessary.
 
The patty melt has come in and out of fashion depending upon dining trends. You’ll frequently find them in diners, especially those that are either styled after 1950s diners or that have been in business since the 50s. In the 1990s, the patty melt got a boost in popularity when several chain fast food restaurants began to offer the sandwiches. To keep this hamburger style manageable and easy to eat on the go, many fast food restaurants use round bread, similar to a bun without the rise. However, if it’s not made with rye bread, regardless of shape, many people consider it a variant of the patty melt and not authentic.
 
Patty melts are said to have inspired the tuna melt, which uses tuna instead of a hamburger patty. Otherwise, they are virtually the same; they are grilled to melt the cheese and can include grilled onions. Tuna melts became popular in the late 40s, and many people preferred them because of the lower fat content in tuna.
 
(Oxford English Dictionary)
melt, n.
N. Amer. With modifying word or words: a sandwich, hamburger, etc., containing or topped with melted cheese.
1956 Diner, Drive-in Sept. 51/1 (caption) Taste-appealing ‘Patty-Melt’—grilled cheese and a hamburger patty.
1970 Nation’s Restaurant News 27 Apr. 15/1 (advt.) Serve the Beef ‘n Cheese Melt!.. Grill it, and the cheese melts over the beef for a flavor combination that’ll start ‘em cheering!
1978 Forbes 17 Apr. 91 All agree a $2 Patty Melt at Denny’s is a bargain.
1989 D. LEAVITT Equal Affections 17 He made tuna melts on seven-grain bread.
1994 Bon Appétit July 82/2 (heading) Open-face Monterey Jack melt. Team the sandwich with tortilla chips and carrot sticks.
 
September 1956, Diner, Drive-In magazine, pg. 51, col. 1:
(Photo caption next to menu and food photo—ed.)
Taste-appealing “Patty-Melt”—grilled cheese and a hamburger patty—score with Scrivner’s customers. Menu merchandising promotes sandwich.
(Menu—ed.)
PUPPY-MELT   FEATURING KRAFT CHEESE 40 cents
Melted Cheese Sandwich with a Grilled Hot Dog
EGG-MELT   45 cents
Grilled Cheese and Scrambled Egg Sandwich
PATTY-MELT   45 cents
Grilled Cheese and a Hamburger Patty
HAM-MELT   55 cents
Grilled Cheese with Virginia Baked Ham
(Col. 2 of text—ed.)
Scrivner’s Drive-Ins in the Los Angeles area spotlight both a “patty-melt” and a king-size “bonus cheeseburger.”
   
An illustrated menu box promotes the “melt” sandwiches “featuring Kraft Cheese” for quality emphasis. To make the patty-melt, two pieces of bread are placed on the griddle, a 3/4-oz. Kraft Ribbon Slice of cheese on…
 
7 July 1960, Long Beach (CA) Press-Telegram, pg. 82?, col. 2 ad:
ORIGINAL PATTY MELT 65c
A TASTE TREAT. 1/4-lb. ground choice beef. Served with melted cheese on home style rye bread sprinkled with chopped onions, with lettuce, carrot curls, kosher pickle and our own special relish.
(L’s Coffee Shop—ed.) 
 
Google Books
Kotch
By Katharine Topkins
New York, NY: McGraw-Hill
1965
Pg. ?:
Anyway he did keep the books for her and she named her patty melt for him, the Jonathanburger.
   
Google Books
New Directions 55
An International Anthology of Prose & Poetry
1966
Pg. 110 (“Style” by Henry H. Roth):
After I ate half my tuna-melt sandwich and drank the good coffee, I announced loudly, “I told Charles I’d stand behind him. “
 
14 March 1966, Tucson (AZ) Daily Citizen, pg. 28, col. 6 ad:
Reubens, Patty Melts, Pork Tenderloin, etc.
(Mickey’s Restaurant—ed.)
   
Serious Eats: A Hamburger Today
The Hamburger Fatty Melt, a Burger with Two Grilled Cheese Sandwiches as Its Bun
Posted by Adam Kuban, October 2, 2008 at 1:00 PM
Update: So many of you asked for bacon on this thing that we just had to oblige. Here is the Bacon Hamburger Fatty Melt. [10/13/2008]
 
What you are looking at, ladies and gents, is what we at A Hamburger Today are calling the Hamburger Fatty Melt. Lemme give you the smack on this meat stack. From top to bottom:
 
Grilled cheese sandwich as bun top
Four-ounce beef patty
Grilled cheese sandwich as bottom bun
 
Got that? It’s a burger with two grilled cheese sandwiches as its bun.
 
I wish our R&D department here at AHT HQ could claim this as the product of our grease-addled minds, but we’re only following through on a burger we heard about via Serious Eats community member Theadob, who mentioned the “Chubby Melt” at the Mossy Creek Cafe in Fisherville, Virginia (burger between two grilled cheese sandwiches, smothered with sautéed onions and mushrooms and topped with Thousand Island dressing).