In a medium skillet, melt 2 tablespoons of the butter over medium low heat. Add the sliced onions and cook slowly until the onions are golden brown and soft, 20 to 25 minutes, stirring occasionally.
While the onions are cooking, In a medium bowl, mix together the ground beef, Worcestershire and salt and pepper to your liking, form into 4 patties.
I use block cheese, Swiss, from New Zealand, excellent cheese, so slice your cheese while the onions are cooking as well. You can slice as thin or as thick as you want, figure about 3 slices per side.
When the onions are done, melt 2 tablespoons of the butter in another skillet over medium heat. Cook the patties on both sides until totally done in the middle, 3 minutes or so each side.
To assemble each sandwich, slice of bread, slices of cheese to cover bread (about 3), hamburger patty, a quarter of the cooked onions, more slices of cheese and another slice of bread.
Using a griddle or a large skillet over medium heat, cast iron would be perfect, melt 2 tablespoons of the butter and grill the sandwiches until golden brown on the first side in the pan. Remove the sandwiches and add the remaining 2 tablespoons butter to the skillet. Return the sandwiches to the skillet, but flip them to the other side. Cook until golden brown and crisp, and until the cheese is melted. Slice in half and serve immediately!