Season burgers with salt and pepper, and cook to desired doneness. While burgers are cooking, caramelize your onion in a sauté pan, with a little oil, salt and pepper. Also butter one side of each piece of rye bread. Once burgers are done, place 4 pieces of rye bread down in a pan, and turn on heat to medium. Add the burgers, grilled onions, and 2 slices of cheese on top of each piece of bread, then top with the other piece of rye, buttered side up. Once, bottom side is browned and crispy, flip over and brown the other side. Add condiments to your own liking and enjoy.