LA Hounder coming Austin
Magnolia Cafe is definitely not fine dining, so you ought to skip it.
I feel I should warn you that "fine dining" is perhaps the least interesting restaurant category in Austin. And you won't find a specifically vegetarian place that does it.
I'd suggest, however, that you try Vespaio. Their menu is very innovative, their execution consistent, and the flavors intense. You'll find vegetarian offerings amongst the pastas, pizzas, antipasti, and appetizers. My favorite dishes are the risotto, the supplì, the fish specials, and the pizzas (not all of which are vegetarian, I know). I've never had a really good dessert at Vespaio, however, and some of their pasta dishes and antipasti are problematic. You can do a search for "Vespaio" on this board to see what other 'hounds like there.
I would also suggest Uchi, but there's no need to try Austin's version of fusion-sushi when you (1) are vegetarian and (2) have access to better versions of same in L.A. Ditto for Wink, which is the only restaurant in town doing ingredient-driven cooking inspired by Alice Waters. Simple food like this has to be perfectly prepared, however, and their kitchen has execution problems. You can do a search for each of these restaurants to see more opinions of them.
I hope this helps.
I didn't want to be the first to say it, but now I'll pipe up and agree with MPH. Magnolia Cafe is definitely NOT what you're looking for. I'm an omnivore and a seeker of low-priced deliciousness, so I don't feel qualified to make recommendations for vegetarians looking for fine dining, but I certainly do feel qualified to recommend against Magnolia Cafe. It is low mediocre, at best.
Thanks so much for the recs Texans! We have to host a luncheon for 23 at Chez Zee and after looking at their menu I was freaked out! (read - is everything breaded and fried???) These sound like great choices - we're in Austin for 5 nights so I might be able to get a few of these in!
I'm a pastry chef in LA so if you ever come here and need recs - please post on the LA Board!
I've never been overly impressed with Chez Zee's menu. It's solid, but nothing exceptional. In my opinion, you won't have a really bad meal there, but you won't have a phenomenal one either. The reason people in Austin go to Chez Zee is the desserts. They are their bread and butter and many are quite exceptional. I'm partial to the Lemon Rosemary cake though I think they are a little heavy on the icing. All the desserts are generally very good.
I agree with Vespaio as a good vegetarian choice. I got the vegetarian plate there and it was better than my husband's carnivore plate. South Congress Cafe is close to it, and they have an excellent menu as well with usually several veggie options.
I've enjoyed Mandola's lately. It's not upscale, more bistro, but excellent Italian when they are on. They offer a wide selection of fresh pastries and gelato as well along with Italian groceries, fresh meats and cheeses.
I also agree with Uchi as a good place to go. I had the tasting menu there about 2 months ago, and I thought it was phenomenal. Probably some of the best cutting edge cuisine in Austin currently. The tasting menu changes from night to night, but standouts when I was there were the seared foie gras paired with vanilla and kumquats, sashimi, green apple sorbet, and the scallops. Uchi definitely is a place to see and be seen, but in typical Austin style it's not overly pretentious IMHO. They have plenty of vegetarian options on the regular menu and I suppose the tasting menu may/may not be vegetarian depending on the night.
I also like the Eastside Cafe. It's more upscale casual, but they specialize in vegetarian (I think they grow many of their own vegetables) and I've consistently had excellent vegetarian options there. The gazpacho soup is very good along with most of their sides (they change daily based on availibility). I've had good luck with their entrees as well, but I've mostly gotten the fish.