A “spaghetteria” is an eating establishment that serves spaghetti. The term might have started in New York City. The New York (NY) Evening Post stated in 1928:
 
“The cafeteria started the craze, then the graceria and many others, but one of the Italian eating places in the Times Square district has captured all honors to date. It has ‘Spaghetteria’ in large letters on its window.”<?i>

 
   
Wiktionary: spaghetteria
Etymology
spaghetti +‎ -eria
Noun
spaghetteria f ‎(plural spaghetterie)
1. spaghetti factory or shop
2. restaurant serving spaghetti dishes (etc)
 
Old Fulton NY Post Cards
30 August 1928, New York (NY) Evening Post, pg. 20, col. 4:
Spaghetteria
The cafeteria started the craze, then the graceria and many others, but one of the Italian eating places in the Times Square district has captured all honors to date. It has “Spaghetteria” in large letters on its window.
 
Old Fulton NY Post Cards
27 November 1937, Rochester (NY) Daily Record, pg. 2, col. 7 ad:
FRANK CUTALI’S
SPAGHETTERIA
SPAGHETTI
“At Its Best”
25c
 
13 April 1941, Boston (MA) Herald, “Let’s Grab a Bite” by Silas Bent, This Week magazine, pg. 6, col. 5:
Frankfurters and hamburgers combined, however, are not the end-all of the quick lunch by any means. New York has had its spaghetteria, and Los Angeles its chowmeinery.
 
Google Books
27 May 1985, New York magazine, pg. 94, col. 2:
Spaghetteria (126 EAST ?TH STREET, 475- 8180) is off the beaten track when it comes to restaurants on Avenue A.
   
Eater SF
What You’ll Eat at Upcoming Nostra Spaghetteria and Bar Mia
by Stefanie Tuder Aug 25, 2015, 5:22p @eatersf
(...)
NOSTRA SPAGHETTERIA
Just like at Plin, the focus is still heavy on fresh, handmade pasta (“It is a spaghetteria,” Alioto said). There are a few Plin holdover favorites, like the egg-filled raviolo ($15) and spaghetti with Grandpa George’s sausage ($16), but there’s also a make-your-own pasta menu (a four-step process that includes choosing your pasta, sauce, vegetables and protein) for $12- $18.