Melt 1 tablespoon of butter in a large skillet (or on a griddle) over medium heat.
Add onions and cook until the onions are soft and golden, stirring often, 20-30 minutes. Remove onions.
Place beef into a large bowl along with salt and pepper, to taste, and the Worcestershire sauce. Mix well.
Form meat into 4 4 ounce patties.
Melt 2 more tablespoons of butter in the skillet. Add the beef patties and cook on both sides until done. Remove meat and any scraps.
Melt 1 more tablespoon of butter in the skillet. Working in batches, brown the bread on one side, while melting cheese on the topside of each slice. Add more butter as needed.
Assemble patty melts by placing burgers on bread and top with onions and remaining bread slice.