Melt 1 tablespoon butter in large skillet over medium-high heat. Add the mushrooms and onion; cook until the mushrooms and onions are dark and caramelized, stirring occasionally.
Spread the remaining butter over 1 side of each slice of bread. Place the bread, buttered side up, on a baking sheet. Broil until golden, watching closely and rotating halfway through, about 3 minutes. Cool. Turn 6 slices over; top with ham, sliced cheese and mushrooms/onions. Top with remaining bread, broiled side up. Spread 2 tablespoons sauce on top, leaving ⅓” plain border. Sprinkle with the Parmesan.
Bake at 450F until cheese inside melts and cheese on top is golden, about 10 minutes.
Adapted slightly from Marlena Spieler for Bon Appetit, March 2011.