Cut 4 1/2-inch thick slices from the center portion of the onion. Divide beef into 4 portions and form each into thin patties. Season patties with salt and pepper.
Heat a skillet over medium-high heat. Cook onion slices for about 2 minutes, then add burger patties and cook until golden brown.
Add cheese slices on top of one onion slice and both burger patties. Stack the burgers and onions, press down, and cook for additional time. Repeat with the remaining ingredients.
In a bowl, combine mayo, ketchup, mustard, and pickle relish. Serve with burgers.