Prepare the low-carb special sauce by whisking the mayonnaise, sugar-free ketchup, unsweetened pickle relish, and white vinegar in a small bowl. Set aside in the refrigerator to let flavors meld.

Divide the ground beef into 4 equal portions and gently roll them into loose balls. Do not overwork the meat to ensure tender patties.

Heat a large cast-iron skillet over medium-high heat until very hot. Place the beef balls into the skillet, spacing them apart so they do not steam.

Use a heavy spatula to firmly smash each meatball flat until it is about 1/4-inch thick. Season generously with the salt and black pepper.

Cook for 2 to 3 minutes until a dark crust forms on the bottom. Flip the patties, add a slice of cheddar cheese to each, and cook for 1 more minute until the cheese is melted and the meat is cooked through.

Assemble the meal in bowls: layer the shredded iceberg lettuce, top with the cheesy smashed patties, add the chopped white onion and dill pickles, then drizzle generously with the prepared creamy sauce.