Stir together mayonnaise, ketchup, mustard, relish, garlic powder, and smoked paprika in a small bowl. Cover and refrigerate until ready to use. This gets better after 30 minutes but works immediately.

Divide the chilled ground beef into 4 equal portions, about 6 ounces each. Loosely shape into balls without packing - the ragged edges create better crust. Keep refrigerated until the pan is screaming hot.

Heat a cast iron skillet or griddle over high heat until it just starts to smoke, about 3-4 minutes. Add 1 tablespoon oil and swirl to coat.

Working in batches to avoid crowding, place beef balls in the pan. Immediately smash each down firmly with a spatula to about 1/4 inch thick - use parchment paper if your spatula sticks. Season with half the salt and pepper. Cook undisturbed until the edges are deeply browned and crispy, about 2-3 minutes.

Flip the patties, scraping up any crispy bits. Season the second side with remaining salt and pepper. Top each with a slice of American cheese. Add a splash of water to the pan and cover briefly to melt the cheese, about 30 seconds. Transfer to a plate and repeat with remaining beef.

Spread house sauce on both halves of the toasted buns. Stack patties on the bottom bun, add top bun, and serve immediately with extra sauce on the side.