Divide the ground beef into 4 equal pieces and roll into balls.
Heat the oil in a cast iron skillet or flat-top grill over high heat. Once the oil is glistening, add half of the onion and cook, stirring often, until browned, about 4 minutes. Transfer to a plate and return the skillet over high heat.
Meanwhile, in a small bowl, combine the mustard, mayonnaise, ketchup, and the remaining onion.
Once the oil is smoking, working one at a time, add a beef ball and top it with a small square of parchment paper. Using a thick spatula, press the beef into a ⅛-inch-thin patty. (The beef may break apart around the edges–that's ok!) Remove the parchment, season the patty with ¼ teaspoon of the salt, and cook until browned, about 2 minutes. Flip the patty, season it with ¼ teaspoon of the salt, and top it with 1 tablespoon of the cooked onion and a slice of cheese. Cook for about 1 minute more, or until the cheese has melted. Transfer the burger to the bottom half of a bun and repeat with the remaining ingredients.
Top each burger with the burger sauce, lettuce, tomato, and top half of a bun. Serve immediately.