I never expected to discover something on my own, let alone in Hyde Park, but our local pan-Asian noodle house is now serving home-made bao for breakfast on the weekend (8-11am). I had two each of Mongolian Beef and Chicken Chow Siu (they're two for $3) and they were really nice. They're not just home-made, they're made to order; so the vegetables in particular are amazingly fresh and the dough was just the right shade of fluffiness.
If you go, I recommend calling ahead (773.684.2801) to place an order, as it takes them about 20 minutes to make, as I found when I went last week while very hungry. I broke the manager's horns a little bit last week when it took so long, and he agreed that they could start them in advance "if we know you're coming." Just tell them you spoke to the obnoxious tall guy with the long hair. They don't seem to have had many takers, so they're not making them in advance.
They have six flavors: BBQ pork, chicken basil, vegetable curry, mongolian beef, chicken chow siu and beef chow siu. The Mongolian beef meat was actually a little dead for me, but it was leavened with some marvelous fresh green that really saved it. The chicken chow siu was really marvelous, with ginger that really sang of being freshly shaved and then just lightly steamed.
I would have sampled more flavors for you, but I didn't know I was going to like them so much I would have to post.

They were also out of a couple flavors by Sunday morning, so apparently Saturday's the better day to go.
Noodles Etc
1333 East 57th Street
Chicago, IL 60637
773.684.2801
http://noodlesetc.com/index.html