Slice the onions thinly. In a large skillet, melt 2 tablespoons of butter over low heat. Add onions and salt, stirring frequently for 30-40 minutes until caramelized.

Mix the ground beef with Worcestershire sauce, salt, and black pepper. Form into patties about ¾ inch thick. Cook in a skillet over medium-high heat for 3-4 minutes per side.

Butter one side of each slice of rye bread. On unbuttered sides, layer cheese, beef patty, caramelized onions, and another slice of cheese. Top with remaining buttered slices.

Grill the sandwich in a skillet over medium heat for about 3-4 minutes per side until cheese melts and bread is golden brown.

Let the grilled Patty Melt rest for a minute, then cut in half and serve hot.