This is no means meant to be the last word on braciole, but a very good one is frequently served at a place called Jimmy's in Forest Park (on Madison next to the brand new Mia Franseca or the former Tutto Tapas).
Jimmy's is truly a bar that also serves excellant homestyle Italian food and killer martini's. Although it is a bar, and can be quite smokey, it is also extremely family friendly. The braciole is a frequent special, but not always available. Still, whatever the specials that day, most everything will be good in an old fashioned way, including the pizza, the fried calarmari and the red sauce.
I often bemoan the over-abundance of places serving the same red sauce/veal parmigiana/fried calamari/chicken vesuvio, etc., but if I had to go to one place for that kind of food, it would probably be Jimmy's.