01 - Set a cast-iron skillet or flat griddle over high heat until the surface is shimmering hot, approximately 3 to 4 minutes.
02 - Divide the ground beef into 4 equal portions and roll each into a loose ball. Avoid packing the meat tightly and do not season at this stage.
03 - Spread butter on the cut sides of each bun. Place them buttered-side down in the skillet and toast until golden brown, about 30 seconds per side. Remove and set aside.
04 - Place the beef balls onto the hot skillet. Immediately press each ball flat using a sturdy metal spatula and a sheet of parchment paper, achieving a thickness of about 1/4 inch. Season the top surface generously with salt and pepper.
05 - Cook the patties for 1 1/2 to 2 minutes until deep golden crust forms on the bottom and juices begin pooling on top. Flip each patty and immediately lay one slice of cheese on top.
06 - Continue cooking for 1 additional minute until the cheese is fully melted and the patty is cooked through.
07 - While the patties cook, combine the mayonnaise, ketchup, yellow mustard, pickle relish, and smoked paprika in a small bowl. Stir until well blended.
08 - Spread the burger sauce on the bottom halves of the toasted buns. Layer each with lettuce, a tomato slice, pickles, and onion rings. Place a cheesy patty on top and cap with the bun. Serve immediately while hot.