Brioche in Austin
Upper Crust makes two fantastic brioche rolls. One is stuffed with dark chocolate and raspberry filling, but I prefer the second option which is stuffed with gruyere cheese. They only make them two days a week - I believe Friday and Saturday. I can't remember what time they come out, but they sell out quickly. You can call and find out about the time.
Whole Foods downtown does make plain brioche rolls. I don't know whether or not they make loaves. The staff is incredibly helpful, though. You should ask the next time you're in there. Someone can probably tell you what the baking schedule is.
And renz was right. Central Market does carry challah. Not all the time, though, I believe. You'll need to ask what their baking schedule is like as well.
There is a Randall's off 2222? (maybe?) and MoPac that supposedly has a big Jewish foods selection inside. I would imagine they might carry challah.
And not that you asked, but Whole Foods downtown has had bread baking classes in the past where you make brioche. They are pretty good classes.
I've also had so much difficult finding brioche (plain) that I ended up trying to make it at home. If you find a good source, please report back.
I checked Whole Foods on Lamar and, at least on the occasions I was there, they didn't have any (although the Arboretum location said that Lamar carried it, it's an infrequent thing or maybe I have bad timing). I check almost every time I go and never have seen it.
Central Market definitely carries challah; I haven't tried it. Whole Foods probably has it as well.
I've asked around to a number of bakeries in Austin and Upper Crust is the only I know that makes brioche. Unfortunately they don't make loaves, but only brioche rolls and huge stuffed brioche. Also, they only make them on the weekend (at least last time I checked). If you find another place, post it and let us know!