In a large bowl, combine the shallots, vinegar, mustard, and tarragon. Slowly add 1⁄3 cup oil while whisking constantly. Season with salt and pepper to taste. Set aside.
Divide cheese into 4 even, round patties and flatten to 1/2'' thick. Coat each with egg, then the breadcrumbs. Repeat with the egg and breadcrumbs. Chill cheese patties for 15 minutes.
Heat remaining oil in a small, non-stick skillet over moderately high heat until hot but not smoking. Cook both sides of the cheese until golden, about 30 seconds per side. Remove and drain on a paper towel.
Toss greens with the salad dressing. Serve salad topped with a warm cheese patty.