Spago, Hatfield's or Lucques?
If you consider Spago, go for the tasting menu. It is phenomenal and included three courses that I would rank on my top 10 of all time. They will accommodate vegetarians; one of our party was vegetarian and her food was excellent.
If you do not do the tasting menu, rave to your vegetarian friend about the agnolloti with truffle. I don't think it is normally part of the tasting menu, but one of my friends began raving about it when we sat down, so they included it as a course in the tasting menu. It was absolutely outstanding. (In the summer, they make the agnolloti with sweet corn, but in the fall, they do it with cauliflower. I was told the sweet corn is better, but I can't imagine a dish better than the cauliflower.)
You will be a winner regardless of the place you select from this list.
All do vegetarian dishes, yet with the wider menu in general, Spago and Lucques might be better options, even though Quinn Hatfield does superb veggie fare.
He just might not have anything you might want to order the night you are there.