"Burger” is a shortened form of “hamburger.” Ben Zimmer of the Oxford English Dictionary found the following early citations from Penn State.
Penn State Collegian
22 July 1926, Penn State Collegian, pg. 1, col. 1:
New Vernacular Reaches Popularity As Jack’s Students Become Greater
State College grows more cosmpolitan daily. Jabbers of French mingle with staccato explosions of Greek at local refreshment parlors. The melodious gutteral [sic] of Spanish passes back and forth across restaurant tables. Native Italian vies with “Pennsylvania Dutch” on Co-op corner. But a new and strange language, about which no textbooks have ever been written, is slowly engaging the attention of an already bewildered populace. Such expressions as “draw one,” “set up,” “burger on one,” “a glass,” “one up,” “cowboy special,” “chip,” will soon become common expressions in the ever swelling collegiate vernacular.
Jack, the short order specialist and chef par excellence at Jerry O’Mahoney’s club diner de luxe, is responsible. He serves, along with his food, a brand new, up-to-date line of chatter which is as amazing as it is puzzling. To him bread and butter isn’t bread and butter, it’s a “set up.” “Draw one” indicates a customer’s desire for a cup of coffee; “a glass” is the wagon king’s nomenclature for a bottle of milk. Perched on the revolving stools before the long lunch counter the amazed customer can only gasp and order a piece of pie out of pure curiosity.
“One cut,” yells Jack. “Comin’ down” replies Bill, his co-partner in slang.
“Two hamburg sandwiches,” orders the undaunted customer.
“Two burgers on one,” the order goes. “Alley Oop,” Bill comes back.
The pie and sandwiches vanish. The customer looks wistfully at Jack. “How much?” he asks.
“Two bits,” says Jack. “Business is pickin’ up; I know my language.”
Penn State Collegian
9 September 1926, Penn State Collegian, pg. 1, col. 5:
“Three Bergers, Draw One, Hat On An Apple"--New Penn State English Course.
A little dictionary expansion is nothing to “Jack” and his worthy assistants who are coining a new type of chatter for Penn State lads and lassies at Jerry O’Mahoney’s “Get-it-quick” Club Diner.
Jack, the boy who’s running the joint for Jerry, is the mint where all the slang is coined. He’s revolutionizing the vernacular of the lunch room. “Burr’ tose,” “pitch-pie,” “ruz-biff,” “scup-cuffy,” and “bowl-zupp” have long since served their turns. They are no more.
At “Jacks” a customer is served not only food but a brand new kind of chatter that leaves him dumb with amazement. To Jack, a plate of beans is
not a plate of beans at all, it is “a thousand.”
“Adam and Eve on a raft,” he yells, and, to the customer’s surprise, Morris, his man Friday, slides up two poached eggs on toast.
In a like manner bread and butter is “a set up,” toast is “angel food,” butter is “a chip” and milk is “a glass,” a hamburg sandwich is “a berger.”