In a large heavy-bottomed skillet, preferably cast-iron, over high heat, melt 2 tablespoons of the butter. As soon as it gets frothy, add the onion slices to the pan and sprinkle with 1/4 teaspoon of and 1/4 teaspoon of pepper. When the onions start to wilt, toss them and cook for about 5 minutes, stirring occasionally.

Reduce the heat to medium and continue to cook the onions, stirring occasionally, until the onions are soft and caramel in color. This will take about 20 minutes.

While the onions are cooking, prepare the burgers. In a large bowl, place the ground beef and season with the rest of the salt and pepper. Using clean hands, mix the seasonings into the beef. Form the seasoned beef into 4 patties and set aside on a platter.

Remove the onions to a bowl. Add 2 tablespoons of butter to the skillet and increase the heat to high. When the butter has melted, add the beef patties to the pan and cook them for about 3 minutes without moving or flattening them to release juices. Turn the patties over and continue to cook for another 2 minutes until medium rare. Remove the patties to a platter and allow them to rest for a few minutes. Turn the heat down to medium under the skillet.

In the same skillet over medium heat, add 1 tablespoon of butter to the drippings in the pan. Place 4 slices of rye bread in the pan. Place a piece of cheese on each slice of bread. Add a patty to the cheese. Spoon equal amounts of onions over the patties. Top with another slice of cheese and then the remaining 4 slices of bread. Using the spatula, lightly press the sandwiches into the pan drippings. You may have to do this in two batches.

After about 1 minute, check that the bottom slice of bread is toasted and golden brown. Carefully turn the sandwiches over. Add the remaining tablespoon of butter to the pan in small portions around the perimeter of the pan. Tilt the pan to move the butter under the sandwiches.

Once the bread is golden brown and crisp on both sides, and the cheese is melted and gooey, transfer the patty melts to a cutting board.