If we're doing institutions/traditions/places that have been around forever and have come to symbolize Chicago to me:
Maxwell Street Express and Jim's Original on Union for Polish & pork chop sandwiches
Beef at Al's followed by Italian lemonade from Mario's across the street, on Taylor (this
is Chicago summer for me)
Lem's
Billy Goat Tavern, on Hubbard and lower Michigan.
Shrimp from Troha's, 4151 W. 26th. (It's been there for 88 years, so I hope it qualifies as an institution.)
Borinquen, birthplace of the jibarito
Hmm...looks like I'm very short at real sit-down restaurants. Well, I guess you can tell my eating habits.

I would also add places like Gibson's, Gale Street Inn, Twin Anchor, Pump Room, etc., to the list. Not necessarily my favorite places by a longshot, but definitely Chicago institutions.