To make the burger sauce- in a blender combine mayonnaise, ketchup, mustard, dill pickle slices, garlic powder, paprika, and cayenne pepper. Blend until smooth. Pour into a bowl, cover, and refrigerate until needed.
Form ground hamburger into 4 patties. Season on both sides with salt and pepper. Heat an outdoor grill to medium-high heat. Grill hamburgers for 5-7 minutes per side or until the internal temperature has reached 160 degrees. During the last few minutes of cooking top each burger with a slice of cheddar cheese. Once the cheese has melted remove from the grill.
To assemble burgers. Place a hamburger patty on the bottom half of a bun. Top with lettuce and tomato. Spread a layer of the burger sauce onto the bottom of the top half of the bun. Place on the top burger.