Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add sliced mushrooms and onion, and sauté 5 minutes or until tender. Drain.
Spread mayonnaise on 1 side of half of bread slices; top with Monterey Jack cheese and tomato. Spoon mushroom mixture evenly over tomato; top with American cheese. Cover with remaining bread slices.
Spread 1 tablespoon butter on 1 side of each sandwich. Cook sandwiches, buttered side down, in a hot nonstick skillet or griddle over medium heat until browned.
Spread remaining 1 tablespoon butter evenly on ungrilled side; turn and cook until lightly browned.