Mix mayonnaise, ketchup, relish, cornichons, onion, Worcestershire sauce, garlic powder, onion powder, and hot sauce in a small bowl; season sauce with salt and pepper.
Do Ahead: Sauce can be made 2 days ahead. Cover and chill.
Prepare grill for medium-high heat. Divide beef into 8 portions (about 6 oz. each). Working with 1 portion at a time, cup your hands around the meat and begin to gently shape it into a round patty, being careful not to pack the meat too tightly. Lightly press down on patty to flatten, then press a slight indentation into the center with your thumb (this will prevent patties from bulging when cooked).
Generously season both sides of patties with salt and grill 5 minutes. Turn and top with cheese. Cover grill and cook until cheese is melted and patties are medium-rare, about 5 minutes more. Transfer to a platter and let rest 5 minutes.
Build burgers with buns, patties, bacon, lettuce, tomato, onion, pickles, and special sauce.