Preheat griddle over medium-high heat. Melt butter then add onions to the griddle. Let cook for 5 minutes before adding the burgers. Season burger patties with salt and pepper. Press burgers firmly onto griddle with the back of a spatula to ensure a good sear. Let cook for 2-3 minutes, or until a nice crust has formed. Flip burgers, and then add 2 slices of cheese to half of the patties. Cook another 2 minutes. Top cheesed patties with uncheesed patties, then remove from grill. Turn heat off on the griddle, then toast buns on it for 30-45 seconds. Top buns with lettuce, tomato, ketchup, mustard, burgers and onions, then serve.