In a small bowl, whisk together the mayonnaise, sweet pickle relish, mustard, vinegar, paprika, onion powder, and garlic powder until smooth. Set aside.

Shred the lettuce until fluffy, chop the onion finely, and have the pickle slices ready. Refrigerate the lettuce to keep it cold.

Heat a large cast-iron skillet over high heat until very hot. Divide the beef into 4 loose balls. Place two balls in the skillet, smash each into a thin patty about 4 inches wide, and season with salt and pepper. Cook without moving for 2 minutes, then flip. Top one patty with a slice of American cheese, cook 1 minute more, and stack the plain patty on top of the cheese. Transfer to a plate and repeat with the remaining beef.

Toast the cut sides of the buns in the skillet fat for 30 seconds. Spread sauce on the bottom bun, add onions and lettuce, place the double patty stack, add pickles, top with more sauce, and close the burger. Serve immediately.