Make the sauce first. Stir together the mayonnaise, ketchup, mustard, chopped pickles, vinegar, smoked paprika, garlic powder, and a pinch of salt. Cover and refrigerate while you cook.

Divide the ground beef into 4 loose balls (about 4 oz each). Don't overwork it and don't season yet.

Melt the butter in a large cast-iron skillet or on a griddle over medium heat and toast the buns cut-side down until golden, about 1 minute. Set aside.

Wipe out the skillet and heat it over high until it's just starting to smoke. With 80/20 beef you won't need any oil.

Place a beef ball in the hot pan and immediately smash it flat with a sturdy spatula (a piece of parchment between the spatula and beef keeps it from sticking). Aim for about 1/4 inch thick and 4 to 5 inches wide, then season the top with salt and pepper.

Cook undisturbed for about 2 minutes, until the edges are lacy and browned and the crust releases easily. Flip once, add a slice of cheese, and cook 30 to 60 seconds more.

Spread special sauce on both toasted bun halves, stack on the patty (or two for a double) plus any onion, pickles, and lettuce, then cap it and serve right away while it's hot and crispy.