short order cook (1891) short order restaurant (1887)

[Barnhart]

OEDs: 1956.

However, in NewspaperARCHIVE.com, in a column of advertisements:

Short Orders.
Having lately added a short order bill of fare also the services of a good
short order cook, I am now prpared to sever meals all hours during the day
or night up to 1 o'clock.  Give me a call.
Hue Singleton, Decatur Morning Review, Oct. 21, 1891, p 4


And, for short order restaurant from the same data base, in a column of
advertisements only some of the type of which is legible.  However, this
line of time is incontestably readable despite the fact that the line
above and the line below are unreadable (to me):

SHORT ORDER RESTAURANT
Colorado Sprins Gazette, Oct. 23, 1887, p 2