Prepare the Beef Patties. Season the ground beef with salt and pepper, then form into patties slightly larger than the bread slices. Cook the patties in a skillet over medium-high heat until they reach your desired level of doneness, typically 3-4 minutes per side for medium-rare.
Caramelize the Onions. In the same skillet, melt butter over medium heat and add thinly sliced onions. Cook, stirring occasionally, until the onions are soft and golden brown, about 15-20 minutes.
Assemble the Patty Melts. Butter one side of each bread slice. Place a slice of Swiss cheese on the unbuttered side of half the bread slices, followed by a cooked beef patty and a spoonful of caramelized onions. Top with another slice of Swiss cheese and the remaining bread slices, buttered side up.
Grill the Sandwiches. Heat a clean skillet or griddle over medium heat. Place the assembled sandwiches in the skillet and cook until the bread is golden brown and the cheese is melted, about 3-4 minutes per side.
Serve and Enjoy. Once the sandwiches are cooked to perfection, remove them from the skillet and let them cool for a minute or two. Slice diagonally and serve hot with Thousand Island Dressing.