Heat a griddle or large cast-iron skillet over medium heat for at least 10 minutes. Meanwhile, combine the mayonnaise, mustard and pickle brine in a small bowl. Combine the onion powder, garlic powder, paprika, 1/2 teaspoon salt and 1/4 teaspoon pepper in a separate small bowl. Set aside.
Loosely form the beef into 4 balls; set aside. Working in batches, lightly brush the griddle with butter and toast the buns cut-side down until lightly browned, 1 to 2 minutes. Transfer to a plate.
Brush the griddle with more butter and add 1 ball of beef. Press firmly with 2 large crisscrossed metal spatulas for exactly 10 seconds, then gently remove the spatulas. Sprinkle the patties with some of the seasoning mixture and continue cooking, undisturbed, until well browned, 1 1/2to 2 minutes. (Start cooking another burger at this point.) Flip the patty, sprinkle with more of the seasoning mixture and top with 1 cheese slice. Continue cooking until the cheese is melted around the edges and the bottom of the burger is browned, about 1 more minute. Repeat with the remaining beef.
Spread the mayonnaise mixture on the bun tops and add some ketchup. Serve the patties on the buns with the pickles, tomato, red onion and lettuce.
Photograph by Ralph Smith