Puree the mayonnaise, ketchup, mustard, pickles and brine, sugar and cayenne in a mini food processor or blender until smooth. Transfer to a small bowl and refrigerate until ready to serve.
Form the beef into 4 pucks, about 2 inches wide and 1 1/2 inches thick. Refrigerate 20 to 30 minutes.
Melt the butter on a griddle or large skillet over medium heat. Open the buns and place open-side down on the griddle. Cook until lightly toasted, about 2 minutes. Remove to a plate.
Wipe off the griddle. Increase the heat to medium high and brush with the vegetable oil. Working in two batches, transfer the patties to the griddle, about 3 inches apart, and press firmly with a large spatula (or two small spatulas) to flatten into very thin patties. Cook, pressing the edges of the patties, about 20 seconds. Season each patty generously with salt and black pepper. Continue cooking until the edges are crisp and the beef is browned, 1 to 1 1/2 more minutes. Flip the patties; season with more salt and black pepper and top each with 1 cheese slice. Continue cooking until the cheese melts around the edges and the burger is cooked through, about 1 more minute.
Spread the mayonnaise sauce on the bun tops. Serve the patties on the buns with the tomatoes and lettuce.
Photograph by Ralph Smith