Slice the Montrachet into 12 (1/2-inch-thick) slices. (The easiest way to slice goat cheese is to use a length of dental floss.) Dip each slice into the beaten egg whites, then the panko bread crumbs, being sure the cheese is thoroughly coated. Place the slices on a rack and chill them for at least 15 minutes.
Melt 1 tablespoon oil and 1 tablespoon butter in a saute pan over medium-high heat until just under smoking. Cook the goat cheese rounds quickly on both sides until nicely browned on the outside but not melted inside. Top each salad with 2 warm rounds and serve.
Toss the salad greens with enough olive oil and balsamic to moisten, then divide them among 6 plates.