In a small bowl, mix together mayonnaise, ketchup, sweet pickle relish, white vinegar, paprika, garlic powder, and a pinch each of salt and pepper until smooth. Set aside in the refrigerator.

Divide the ground beef into four equal portions (about 4 ounces each) and gently form into loose balls. Don't overwork or pack the meat tightly.

Heat a large 12-inch cast-iron skillet or griddle over high heat until smoking hot, about 4 minutes. Place the beef balls on the hot skillet and immediately press down hard with a heavy-duty metal spatula using your full body weight for 10-15 seconds until the patties are about 1/4-inch thick with lacy, irregular edges. Season with salt and pepper. If your skillet is smaller, work in two batches to avoid crowding.

Cook undisturbed for 2 minutes until edges are dark mahogany brown and crispy with a distinct crust forming. The meat should release naturally from the pan. If patty sticks when you try to flip it, cook another 30 seconds. Flip the patties, top each with a slice of cheddar cheese, and cook for 60-90 seconds until cheese is fully melted, bubbling, and starting to brown at the edges.

Remove patties from heat and let rest for 1 minute, then roughly chop into 1-2 inch pieces.

Divide chopped romaine among four bowls as the base. Top with cherry tomatoes, red onion, and pickles. Add chopped cheeseburger pieces and drizzle with special sauce. Serve immediately.