These are nice suggestions. What I have in mind however, is a bit more of restaurant and not a cafeteria. Kitchen market is close to what I am seeking. I had not thought of them. I will try Despana. ALthough, I am a big fan of Kalustyan, it is too much of a cafetieria set-up. Perhaps, what I seek is very rare. I once went to a place in Queens (I can't remeber the name) that was an Italian market duing the day and a white table cloth restaurant at night. The house specialty was carbonara that was tossed in a hollowed out wheel of parmesan cheese. I was told a wine shop at the South St. seaport area with an Italian name was doing small bites and was a wine/fine food shop as well but I don't know the name. Any other suggestions?
You should check out Despana on Broome Street.
They have delightful bocadillo sandwiches with velvety mellow Spanish ham and if you go during lunch or later, they have really nice olive oils and spreads set out to taste.
And for Mexican items there's Kitchen Market:
It's on Eighth Ave and run by the Bright Food people.
(no food here, just my photoblog)