In a small bowl, combine the mayonnaise, pickle relish, vinegar, garlic powder, onion powder, and paprika. Chill in the refrigerator several hours before serving.
Season the ground beef with salt and pepper (or other seasonings as desired) and place on a cutting board. Place a sheet of plastic wrap over the beef and roll out to about 1/4" thickness. Cut the beef into 8 round, thin patties using a biscuit cutter or the top of a cup, and cook the patties on a grill or in a skillet until cooked through.
To assemble the burgers:
Add the following ingredients, working from the bottom up: bottom bun, special sauce, onion, lettuce, beef patty, cheese, pickles, bottom bun, special sauce, onion, lettuce, beef patty, pickles, top bun.