I don't think there is one answer to your question. Every restaurant makes a choice about what it thinks a high end bun is. I would personally say either a brioche bun or an onion roll.
I think the key is that the bread is fresh, it's different from a white bread bun, and it gets a little TLC itself.
I would choose one of the two buns I did, because there is an obvious taste/texture difference between them and a standard bun. Also, what I meant by TLC is that most places just treat the bun like something to keep the burger grease off your fingers.
You'd be surprise at how it changes the bun if you put a thin coat of butter on it and slap it on the grill/flat top to crisp up. Steaming the bun also really ramps it up in my opinion. We've started steaming our plain white hotdog and hamburger buns and they're awesome. Another technique that's pretty much exclusive to the flat top, unless you have an add-on for your grill, is cooking the patty and/or bun on top of a pile of chopped onions. The onions caramelize, but the onion juices that cook out, steam up and get inside the bun to a certain extent.
So I think the choice of bread is secondary to what you do with it to jazz it up, because to be honest, a high end hamburger is more about the person eating it's perception of it being high end. I've eaten high end hamburgers cheapo hamburgers. I don't taste enough of a difference in the actual meat and bread to make me wanna pay a high price. Typically, the difference is in any seasonings, sauces, or preparations they do with it.