Diner lingo shortens a “grilled American cheese sandwich” to “GAC.” The term “GAC” is pronounced and often written as “Jack.” A “Jack” with bacon is called a ‘Jack Benny” or a “Jack Back” (“Jackback”) or a “GACB.”
 
The term “GAC” for “grilled American cheese” was printed in Hearst’s International Combined with Cosmopolitan (New York, NY) in September 1949, and in a 1967 abbreviations dictionary. “Jack Back” is cited from at least 1973, “Jack Benny” is cited from 1977, and “GACB” is cited from 2009. All of these terms have very few recorded citations, however, so the amount of actual restaurant usage is in question.
   
     
Wikipedia: Diner lingo
GAC: Grilled American cheese sandwich. This was also called “jack” (from the pronunciation of “GAC”)
(...)
Jack Benny: (named after Jack Benny, the comedian) cheese with bacon
 
Wikipedia: Cheese sandwich
Grilled cheese sandwich
A grilled cheese sandwich is a grilled (actually fried) or broiled sandwich consisting of two slices of bread (usually buttered) with cheese melted in between, sometimes combined with an additional ingredient such as peppers, tomatoes or onions. It is a simple variation on the normal cheese sandwich, and is sometimes known as a cheese sandwich, without any qualification. A variety of different names denote the same sandwich, such as cheese toastie or toasted cheese sandwich; in England and New Zealand it goes by toastie, in Australia it is commonly called a cheese jaffle, and in the Netherlands it is called a tosti.
   
Wikipedia: Jack Benny
Jack Benny (February 14, 1894 – December 26, 1974), born Benjamin Kubelsky, was an American comedian, vaudevillian, and actor for radio, television, and film. Widely recognized as one of the leading American entertainers of the 20th century, Benny played the role of someone comically “tight” with his money, insisting on remaining 39 years old despite his actual age, and often playing the violin badly.
 
Benny was known for his comic timing and his ability to get laughs with either a pregnant pause or a single expression, such as his signature exasperated “Well!” His radio and television programs, tremendously popular in the 1930s, 1940s, 1950s, and 1960s were a foundational influence on the situation comedy.
         
27 February 1934, Casper (WY) Tribune-Herald, “Students Give New Meaning to the Alphabet,” pg. 6, col. 7:
LOS ANGELES, Feb. 27.—(AP)—(...) It’s the new vernacular of the work-your-way students who wait on the trade at the Westwood cafeteria of the University of California at Los Angeles and who unofficially constitute the They-Don’t-Speak-Our-Language-Association.
 
The waiter’s shout of “JD” means Jersey dark—a chocolate milk shake to you. “OJ” is orange juice, “SO” limeade (squeeze one), and “CO” lemonade (cut one). “GS” is a peanut butter sandwich (goober san), and “AC” American cheese sandwich.
 
17 April 1949, Sunday News-Democrat (Tallahassee, FL), “Jaunty Jargon of Soda Fountain Meaningful: ‘Jerks” Have a Word For It—or Number!” by Steve Yates, sec. 2, pg. 9, col. 2:
GAC—Grilled cheese sandwich.
 
September 1949, Hearst’s International Combined with Cosmopolitan (New York, NY), “Gimme a 51 and a Parlez-Vous,” pg. 82, col. 3:
Here’s a list which may help you solve some of the mystic language which is shouted behind the soda fountain:
(...)
6. G.A.C. (Grilled American cheese)
   
9 August 1958, St. Petersburg (FL) Times, “Have A Lingo All Their Own” by Nancy Osgood, pg. 4-B, col. 4:
The newcomer to the profession is considered by his fellows a rank amateur who “throws a monkey wrench into the machinery” until he can remember that a grilled American cheese sandwich is not that at all—but simply a “gack” or less briefly, a G-A-C.
 
Google Books
Abbreviations Dictionary
By Ralph De Sola
New York, NY: Meredith Press
1967
Pg. 115:
gac: grilled American cheese (sandwich)
     
Google Books
Harper’s Magazine
v. 247
1973
Pg. 7:
jack back — grilled American cheese with bacon
   
Google News Archive
12 December 1977, Village Voice (NY), “Counter Code,” pg. 60, cols. 3-4:
Jack: Literal origin is GAC—grilled American cheese.
Jack Benny: Grilled American cheese with bacon.
 
Google Books
Listening to America:
An illustrated history of words and phrases from our lively and splendid past
By Stuart Berg Flexner
New York, NY: Simon & Schuster
1982
Pg. 474:
GAC, a grilled .American cheese sandwich, from the initials; also Jack, from pronouncing GAC. A Jack Benny is a Jack with bacon, the B of Benny standing for bacon and the whole term using the name of comedian Jack Benny, whose popular Sunday evening…
       
19 September 1990, Newsday (Long Island, NY), “The Diner” by Sylvia Carter:
A jack - grilled American cheese. Jackback - grilled cheese with bacon.
 
20 July 1992, Lexington (KY) Herald-Leader, “Short-order joints long on colorful vocabulary,” pg. B1:
And a grilled cheese sandwich with bacon was called “a Jack Benny.”
   
Google Groups: misc.writing.screenplays
Newsgroups: misc.writing.screenplays
From: “dti”

Date: Mon, 18 Jun 2001 20:33:16 GMT
Local: Mon, Jun 18 2001 3:33 pm
Subject: Re: Soda Jerk slang—anybody got a source?
 
Here are some other terms for you:
Rye bread = “whiskey” (easier to discern whiskey from white than rye from white, when you’re calling to the cook in a noisy kitchen)
Rye toast = “whiskey down” (any kind of toast is “down”)
A grilled cheese = “Jack” (which is a pronunciation of its abbreviation, GAC which stands for Grilled American Cheese)
Grilled cheese with tomato = “Jack Tommy”
Grilled cheese with bacon = “Jack Back”
Grilled cheese with bacon and tomato = “Jack Back Tommy”
Grilled Swiss Cheese = “Jack Swiss”
Grilled Swiss Cheese with Tomato and Bacon on Rye = “Jack Swiss Back Tommy Whiskey”
Pancakes = “Stack”
Pancakes with Bacon = “Stack Back”
       
26 September 2001, Beaumont (TX) Enterprise, “I need a groundhog with bullets on wheels…”:
A grilled cheese is a GAC (grilled American cheese), pronounced “Jack.” A GAC with bacon is a “Jack Benny.” 
   
Google Books
Sundae Best:
A history of soda fountains
By Anne Cooper Funderburg
Bowling Green, OH: Bowling Green State University Popular Press
2002
Pg. 169:
JACK: Grilled cheese sandwich.
JACK BACK: Grilled cheese and bacon sandwich
   
Google Books
Norman Bereft
By Nicholas D Brown
AuthorHouse
2009
Pg. 96:
Norm attacked his GACB, a grilled American cheese with bacon sandwich.