In a medium bowl, whisk all of the sauce ingredients together. Cover and chill.
PREPARE THE BURGERS:
Wrap the bottom of a Dutch oven with foil. Divide the ground beef into 4 lightly packed balls. Heat oil in a 12-inch skillet over high heat until just smoking. Transfer 2 beef balls to the skillet. Using the prepared pot, press the patties until they are about 4-inches in diameter; set the pot aside. Sprinkle the patties with salt and pepper. Cook the burgers, without moving, until well-browned on the first side, 60 to 90 seconds. Flip the patties and top with 1 slice of cheese. Cover and continue to cook until browned on the second side, about 60 seconds. Transfer the burgers to a platter and cover to keep warm. Pour off all but 1 teaspoon of fat from the skillet and repeat with the remaining 2 beef balls and 2 slices of cheese.
Spread each bun with 2 tablespoons of burger sauce. Serve the burgers on buns, topped with lettuce, tomato, onion and pickles. Pass extra sauce separately.