Piazzale Michelangelo doesn't have a lot in the immediate vicinity (you'll see why when you get there, don't spoil the view with a google search first). Anything up there I'd consider to be fairly touristy - that means likely open all of August, but also not particularly great.

The closest place I have pinned is La Leggenda dei Frati: http://www.laleggendadeifrati.it/

It's nice (and expensive). I can't tell you whether they'll be open in August, but give them a call or email them. The place itself is in Villa Bardini, pretty cool setting for a restaurant.

If you're looking for something a little more casual for dinner, I'd make the walk back across the river to Del Fagioli (about 20 minutes walking) - these guys probably will close for a week in August, so call or email to find out which week. Or for something lighter (or heavier on wine), over to Enoteca Pitti Gola e Cantina. To my knowledge, Pitti Gola is open all of August this year. They've gone full tourist (everything is in English), but the quality hasn't dropped off.