In a large skillet over medium heat, melt 1 tablespoon of butter. Add the sliced onions and cook slowly until deeply caramelized, about 15-20 minutes. Remove from skillet. Add another tablespoon of butter and the sliced mushrooms. Cook until well-browned, 10-12 minutes. Season with salt and pepper and remove from the skillet.

Form the ground beef into two thin patties, slightly larger than your bread slices. Season both sides with salt and pepper. Cook in the same skillet over medium-high heat for 3-4 minutes per side, until a brown crust forms.

Spread the remaining 2 tablespoons of softened butter on one side of each slice of bread. On the un-buttered side of two slices, layer one slice of Swiss cheese, a cooked patty, half of the mushroom and onion mixture, and a second slice of Swiss cheese.

Place the assembled sandwiches in the skillet and grill over medium heat for 3-5 minutes per side. The bread should be golden brown and crispy, and the cheese should be fully melted. Press down gently with a spatula while grilling.

Remove the melts from the skillet and let them rest for one minute before slicing in half. Serve immediately.