In a small bowl, combine mayonnaise, ketchup, dill pickle relish, yellow mustard, white wine vinegar, paprika, garlic powder, onion powder, sugar, and salt. Whisk until smooth and well combined. Taste and adjust seasoning. Cover and refrigerate for at least 30 minutes to let flavors meld (can be made up to 48 hours ahead).

Burger

Form patties

Divide the 1 lb cold ground beef into four equal portions (about 4 oz each). Gently shape each portion into a loose ball, then press into thin patties about 4 inches wide and 1/2 inch thick. Do not overwork the meat. Season both sides of each patty with a light sprinkle of the 1 tsp salt and 1/2 tsp black pepper.

Preheat cooking surface

Heat a large cast-iron skillet or griddle over medium-high heat until very hot (about 5 minutes). Lightly oil the surface if it is not well seasoned.

Cook patties

Place patties on the hot surface without crowding. Cook 2 to 3 minutes on the first side, pressing gently with a spatula to ensure even contact (this helps form a good crust). Flip once and immediately place 1 slice of American cheese on each patty. Cook an additional 1 to 2 minutes until cheese is melted and internal temperature reaches 160°F (71°C). Remove patties to a plate and tent loosely with foil while toasting buns.

Butter and toast buns

Spread softened butter on the cut faces of the 4 buns. Place buns butter-side down on the hot skillet or griddle and toast until golden brown, about 1 to 2 minutes. Remove and set aside.

Assembly & Toppings

Prep toppings

While patties cook, separate lettuce leaves, slice tomato into 4 thin rounds, and trim onion slices into rings. If desired, briefly caramelize onion by cooking rings on the skillet for 1–2 minutes until slightly softened (optional).

Assemble burgers

Spread about 1 tablespoon of the special sauce on the bottom half of each toasted bun. Place a lettuce leaf on the sauce, then a cheese-topped patty. Top the patty with 2 pickle slices, a tomato slice, and a few onion rings. Spread about 1/2 teaspoon yellow mustard on the top bun, close the burger, and serve immediately.