Grill sliced onions in butter until tender. Spread butter on bread and brown, buttered side down, in skillet. When toasted, turn bread over and place a slice of Monterey Jack cheese on one slice of bread and a slice of Mozzarella cheese on the other. Put a beef patty on one side; top with grilled onions, then top with other side of bread and cheese. Flip sandwich as when making grilled cheese to completely heat cheese.