Mix mayonnaise, ketchup, and pickle relish in a small bowl until smooth, then refrigerate while preparing the burgers.
Divide the ground beef into 4 equal portions, shape into patties slightly larger than the buns, press a shallow indent in the center of each with your thumb, and season both sides with salt and black pepper.
Heat a cast iron skillet or griddle over medium-high heat, cook the patties for 3 to 4 minutes per side without pressing, flipping once, until they reach 160°F / 71°C.
Place one cheese slice on each patty during the last 60 seconds of cooking and cover the pan to melt the cheese.
Toast the buns cut-side down in the same pan for 1 to 2 minutes until golden.
Spread the sauce on both bun halves, then layer the bottom bun with lettuce, tomato, onion, the patty with melted cheese, and the top bun.