Cut 4 1/2-inch thick slices from the center portion of the onion and discard the smaller ends for another use.

Divide beef into 4 portions and form each portion into a very thin patty. Season patties with salt and pepper.

Heat a large skillet over medium-high heat. Working in batches, add 2 onion slices to the pan and cook, undisturbed, about 2 minutes.

Add 2 patties and continue to cook until golden brown, 2 to 3 minutes. Flip onion and burger patties and continue to cook until golden brown, pressing the burger as it cooks.

Add 2 cheese slices to the top of one onion slice and both burger patties.

Place one burger on the cheese-topped onion slice and top with the second burger patty. Place the remaining onion slice over the burger and press down.

Flip burger and cook 1 to 2 more minutes. Remove from the pan and set aside. Repeat with remaining onions, burger patties, and cheese.