Heat a cast-iron griddle over medium heat until hot.
Spread 2 tablespoons of the mayonnaise equally on the split sides of the buns and griddle mayo-side down until golden brown, 1 to 2 minutes. Set aside.
Combine the ketchup and mustard with the remaining 1 tablespoon mayonnaise in a small bowl and stir until just combined. Spoon the sauce onto the top buns.
Heat a large cast-iron skillet over medium-high heat. Season the beef with salt and pepper. Divide into 4 equal patties and transfer to the skillet. Cook for 4 to 5 minutes. Add the butter, flip the patties, then remove from the heat (or bake in a 500 degree F oven until the patties are cooked to your desired temperature). Top each patty with a slice of cheese and let it melt.
Transfer the patties to the bottom buns. Top with the pickles and chopped onion, as desired. Close the burgers and serve hot.