In a medium skillet, melt 2 tablespoons of the butter over medium-low heat. Throw in the sliced onions and cook slowly until the onions are golden brown and soft, 20 to 25 minutes, stirring occasionally.
In a medium bowl, mix together the ground beef, Worcestershire and some salt and pepper. Form into 4 patties.
Melt 2 tablespoons of the butter in a separate skillet over medium heat. Cook the patties on both sides until totally done in the middle.
Assemble the patty melts this way: Slice of bread, slice of cheese, hamburger patty, a quarter of the cooked onions, another slice of cheese and another slice of bread.
On a clean griddle or in a large skillet over medium heat, melt 2 tablespoons of the butter and grill the sandwiches until golden brown. Remove the sandwiches and add the remaining 2 tablespoons butter to the skillet. Return the sandwiches to the skillet, flipping them to the other side. Cook until golden brown and crisp, and until the cheese is melted. Slice in half and serve immediately!