I don't know where you will be staying but Ann Sathers is famous for breakfast. The cinnamon rolls are amazing. For deep dish I think Pequods is better than Gino's East but Gino's East may be more conveniently located.
You also can't go wrong with any of Rick Bayless's restaurants. Frontera Grill, Topolobombo(fancier), Xoco. I love Frontera Grill but it has been awhile since I've been there.
Girl and the Goat is fantastic. I don't know ir it is still as difficult to get into.