Mix the ground beef in a large mixing bowl with the salt. Roll into 8 even balls and smash them into flat patties, about ½ inch thick, with your hands. Set aside.
Place patties in the air fryer basket, and top with a slice of butter. Air Fry at 400°F / 205°C for 12-14 Minutes, until golden brown and crispy on the edges. Top each patty with a slice of American cheese and Air Fry at 400°F / 205°C for another 3-5 Minutes, until the cheese has melted. Stack patties into towers of two a piece, creating four stacks and set aside.
Air Fry the hamburger buns at 400°F / 205°C for 2 Minutes, until warmed.
Meanwhile, make the burger sauce. Combine the ketchup, yellow mustard, mayonnaise, and sweet relish in a mixing bowl. Spread the burger sauce on each top and bottom bun.
Layer the shredded lettuce on the lower bun, then a few tomato slices, onion slices, one of the cheesy burger stacks. Finish with the top bun. Enjoy!