First, you may want to define what "classic" means to you. There are distinct differences between "bistro" cuisine--places like Chez Moi, Le Sardine, Le Bouchon and a few other well-regarded spots like, Maude's, Bistro Campagne, Kiki's Bistro and Mon Ami Gabi v. the more formal "haute cuisine" that you'll find at places like les Nomades, Everest (though Everest, actually, isn't true haute cuisine but more of a very elevated country cuisine with an emphasis on Alsace), Tallgrass out in Lockport, Michael in Winnetka and, finally, some contemporary takes on haute cuisine like Brindille and Tru.
It's a pretty wide landscape--though one with an equally wide variation in pricing. Interestingly, the Blanchard looks to be kind of a hybrid between all three--definitely some formal elements (Dover Sole Meuniere), along with some of the more common bistro menu staples (Steak Frites) and the contemporary (four different preps of Fois Gras with non-traditional ingredients, such as apricot gelee' and candied lavender). I haven't been yet--kinda put off by some of the service issues that folks have raised.
Hope you'll report back on your exploration--sounds like fun!
"Knowledge is knowing a tomato is a fruit; wisdom is not putting it in a fruit salad." Miles Kington