Shake Shack started in 2001 as a lunchtime hot dog cart run by the kitchen of Michelin-starred restaurant Eleven Madison Park. The cart was supposed to be temporary, but demand was so huge that in 2004 owner Danny Meyer opened a kiosk in Madison Square Park and expanded the menu to include burgers and shakes. There are now 133 other locations — as far away as Tokyo and Dubai — along with a new cookbook, Shake Shack: Recipes & Stories.
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.