In a large skillet, melt 2 tablespoons of butter over medium-low heat. Add sliced onions and cook them slowly until golden and soft, about 30 minutes. Stir occasionally for even caramelization.

In a mixing bowl, combine the mayonnaise, ketchup, yellow mustard, garlic powder, and ground black pepper. Stir until smooth, then set aside.

Heat a cast iron skillet over medium-high heat and add 2 tablespoons of butter. Cook the patties according to package instructions. Once cooked, remove them and clean the skillet.

Spread the prepared sauce on one side of each slice of bread. On four slices, layer caramelized onions, a patty, both types of cheese, then top with remaining slices.

In the same skillet, add more butter and grill the sandwiches until golden brown and crispy, flipping as needed. Serve immediately.