In a large skillet or cast-iron pan, melt 1 tablespoon of butter over medium-low heat. Add the thinly sliced onions and a pinch of salt. Cook slowly, stirring occasionally, for 20-30 minutes, or until the onions are deeply golden brown and very tender. Once caramelized, remove the onions from the skillet and set aside.

Divide the ground beef into four equal portions and form them into oval-shaped patties, slightly larger than your slices of rye bread. Season both sides generously with salt and freshly ground black pepper. In the same skillet (or a separate one if you prefer), increase the heat to medium. Cook the patties for 3-5 minutes per side, or until desired doneness is reached. Remove the cooked patties from the skillet and set aside.

Wipe the skillet clean if necessary. Melt the remaining 1 tablespoon of butter over medium heat. Lightly spread one side of each rye bread slice with mayonnaise or Thousand Island dressing (if using). Place four slices of bread, buttered-side down, in the skillet. Top each slice with one slice of Swiss cheese, then a cooked beef patty, a generous spoonful of caramelized onions, and another slice of Swiss cheese. Top with the remaining four slices of rye bread, buttered-side up.

Grill the Patty Melts for 3-4 minutes per side, or until the bread is golden brown and toasted, and the cheese is completely melted and gooey. You may need to press down gently with a spatula to ensure good contact with the pan and even toasting. Once both sides are golden and the cheese is melted, remove from the skillet. Cut in half and serve immediately.