It's a stretch to call Au Cheval a diner as it bills itself but it is truly a unique concept. Not open for breakfast and more of a late night/dinner spot. It takes a twist on the old school and uses high end ingredients with a chef driven menu. This place is fantastic by the way.

Best eggplant parm is at Club Lago in River North. Hands down.

Interesting...Volare is one of my favorite go to Italian places in the area.

this may be a little late but Quartino and Osterio Via Stato sound like great suggestions for you. Quartino serves small plate Italian and some very good pizza. Osteria is Italian as well.

As for Big Star, probably not the best suggestion with kids. I've never seen one there. They do check everyones ID's prior to walking in.

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Quartino
626 N. State Street, Chicago, IL 60654

Crust in Bucktown has a good selection of local and regional beers as well as some pretty good think crust pizza.

Also just ate at State and Lake at the Wit hotel. Large selection of beers on tap and more in bottles. Decent food as well.

Regarding a grocery store near by, go to Bockwinkels. It's located on Stetson Ave just a block or two from your hotel.

Regarding dinner near by, as mentioned by another person, I highly recommend the Gage and they accept reservations. Tavern at the Park is pretty good too and located on Randolph St near Michigan Ave.

Tavern at the Park is a block away on Randolph St attached to Pru Plaza. Good burgers and hearty food. It's also the same owners as Keefers, a pretty good steak house not too far away.