Here are a bunch of recommendations that are worth checking out AND are conveniently within about a 3-minute walk of the hotel.
Atwood Cafe is the restaurant in the Burnham itself, and it is very good indeed, for anything from dinner to Sunday brunch. It features contemporary American cuisine, big stuffed seating, and the kind of service you would expect in a luxury hotel.
Pizano's on Madison is a couple of blocks southeast of the hotel and offers an authentic version of our delicious local specialty of deep-dish pizza. If you don't want to wait 30+ minutes while seated for your pizza to bake, you can phone ahead with your pizza order.
Petterino's is a steakhouse a block north of the hotel. In addition to steaks for lunch and dinner, they do a nice buffet Sunday brunch.
For snacking, within a block or two of the hotel are a couple of locations of
Garrett's Popcorn, featuring freshly-made caramel popcorn (with or without pecans or cashews), cheese popcorn, and the "Chicago mix" of the two.
Toni's is an excellent European-style bakery a block east of the hotel. And
Intelligentsia is our premier local coffee roaster, with a cafe on Randolph a couple blocks northeast of the hotel.