Caramelize onions. Melt 2 tablespoons of the butter in a large skillet set over high heat. When it foams, add the onions and sprinkle with salt. Do not stir immediately. Wait 1 minute, then begin to stir frequently over high heat for 5 minutes or so, or until the onions have released some liquid and started to become translucent.
Reduce the heat to medium, and cook, stirring often, for what will seem like a very long time, until the onions are fully melted and dark golden brown, approximately 30 to 40 minutes. Remove onions, and set them and the skillet aside. (You can caramelize the onions a day ahead of time and reheat slowly when you’re ready to cook.)
Cook the burgers. Add the remaining butter to the skillet in which you cooked the onions, and place it over medium heat. Gently divide the ground beef into 4 small piles, and then lightly form these into patties that are like flattened meatballs, roughly the shape of a slice of rye bread. Season aggressively with salt and pepper.
Increase the heat under skillet to high. Put the patties into the skillet with plenty of distance between them, and allow them to cook, without moving, for approximately 2 minutes. Use a spatula to turn the patties over, and continue to cook for another 2 to 2 1/2 minutes, then remove the patties from the skillet, and allow them to rest. (They will at this point be rare inside.)
Decrease the heat below the skillet to medium, and put four pieces of rye bread into the butter and fat that remains from cooking the patties. Top each piece with a slice of cheese, then a patty, then some of the caramelized onions, and finally another slice of cheese and another slice of rye. Use the spatula to press down on the packages, and after about a minute or so, carefully turn over each patty to begin to brown the other side. (You may need to add the additional butter.) Cook until the cheese is fully melted and the bread is golden brown and crisp on both sides. Slice in half before serving.