Homemade Shake Shack Burger

Adapted from J. Kenji López-Alt, Serious Eats [1]

For the shack sauce:
1/2 cup mayonnaise
1 tablespoon ketchup
1 tablespoon yellow mustard
4 slices kosher dill pickles
1/4 teaspoon garlic powder
1/4 teaspoon paprika
Pinch cayenne pepper

For the burgers:
8 ounces ground beef sirloin
4 ounces ground beef chuck
4 ounces ground beef brisket
2 tablespoons butter, at room temperature
4 potato burger buns, such as Martin's Sandwich Rolls
4 tablespoons shack sauce
4 leaves of green-leaf lettuce
8 slices ripe plum tomatoes
1/2 teaspoon vegetable oil
Kosher salt
Black pepper, freshly ground
4 slices yellow American cheese

Combine mayonnaise, ketchup, mustard, pickles, garlic powder, paprika, and cayenne in a food processor and process until smooth. Transfer to a squeeze bottle. (It makes about 3/4 cup sauce.)
Gently mix together the sirloin, chuck, and brisket in a mixing bowl until just combined. Form into 4 patties, each about 2 inches tall and 2 1/2 inches wide. Refrigerate until ready to cook.
Split open buns and spread with a thin coating of butter, then toast in a skillet until brown, about 1 minute. Using a squirt bottle, drizzle about 1 tablespoon shake sauce on the top half of each bun. Place 1 lettuce leaf and 2 slices of tomato on top half of each bun.
Add oil to a large, heavy-bottomed skillet, then heat over medium-high heat until just beginning to smoke. Generously season beef patties on top side with salt and pepper, then transfer, seasoned side down, to the hot skillet. Use a heavy, flat spatula to press down and lightly flatten the beef patties. Season top-facing side with salt and pepper. Cook until a crisp brown crust has formed, about 2 minutes.
Carefully loosen any crusty parts of the burger patties from the skillet, and flip over. Top each patty with 1 slice of American cheese and cook until the cheese has melted, about 1 minute longer. Transfer patties to burger bun bottoms, top with the the top bun (including tomatoes and lettuce), and serve hot.