Will be in town for the Housewares show. If you could eat in one restaurant, which would it be? Must be vegetarian friendly.
I think it depends on whether you want to see (a) what a high-end restaurant can do with their vegetarian tasting menu, or (b) what a high-end restaurant dedicated to vegetarians can do.
a) Our very top haute cuisine restaurants - Alinea, Everest, Charlie Trotter's, Avenues, TRU, Spiaggia - serve multi-course tasting menus, and can provide a tasting menu to accommodate vegetarians in your party. Any of these can do so. Keep in mind that these are expensive ($200-300+/pp) and dressy places. Alinea is the "creme de la creme" of this group, although also the most expensive; Everest is a bit of a bargain.
b) Green Zebra - but ONLY if your group consists solely or primarily of vegetarians. Acclaimed chef Shawn McClain owns/runs three restaurants here: Spring, which emphasizes seafood, Custom House, which emphasizes meats, and Green Zebra, which is almost entirely vegetarian (they often have a single poultry dish and otherwise the entire menu is vegetarian). Meat and fish lovers may not be thrilled.
If you're looking for other "casual fine dining" restaurants that do a very nice job for vegetarians without the high price of the "haute cuisine" group, I recommend North Pond and Lula, both of which emphasize local and seasonal ingredients and have vegetarian options on the menu. If you are not looking to spend a fortune, and your group consists of meat-eaters with one or two vegetarians, these are great choices (particularly North Pond, for the exquisite setting in the middle of the park).
You can get a fine meal at any of these places, including for any vegetarians in your party. But since you wanted ONE recommendation, it would be Alinea, assuming money is no object. It's simply the "best place in town". (If price IS a consideration, then Green Zebra if the group is mostly vegetarians, otherwise North Pond.)
Wherever you go (except Green Zebra, obviously), make sure to let the restaurant know how many vegetarians are in your party, and what kind (e.g. vegan, etc), when making your reservation.
Alinea - www.alinearestaurant.com
Everest - www.everestrestaurant.com
Charlie Trotter's - www.charlietrotters.com/restaurant
Avenues - www.peninsula.com
TRU - www.trurestaurant.com
Spiaggia - www.spiaggiarestaurant.com
Green Zebra - www.greenzebrachicago.com
Spring - www.springrestaurant.net
Custom House - www.customhouse.cc
North Pond - www.northpondrestaurant.com
Lula Cafe - www.lulacafe.com