In a skillet over medium heat, melt 1 tablespoon of butter. Add the sliced onions and cook, stirring occasionally, until they are caramelized and golden brown, about 10-15 minutes. If using, stir in Worcestershire sauce during the last minute of cooking. Season with salt and pepper to taste. Remove the onions from the skillet and set aside.
In the same skillet, add the ground beef and season with salt and pepper. Cook until browned and cooked through, about 5-7 minutes. Drain any excess fat if necessary.
To assemble the patty melts, place a slice of rye bread in the skillet over medium heat. Layer with a slice of cheese, half of the cooked beef, half of the caramelized onions, and another slice of cheese. Top with another slice of rye bread.
Cook until the bottom is golden brown, about 3-4 minutes. Carefully flip the sandwich and add the remaining butter to the skillet. Cook until the second side is golden brown and the cheese is melted, about 3-4 minutes more.
Remove from the skillet, slice in half, and serve immediately.