Divide the beef into 8 equal portions and press each between parchment into a thin 5-inch round.
Place one compact stack of quartered cheese in the center of 4 rounds, leaving a 1-inch bare border.
Cover with the remaining beef rounds and pinch the edges together firmly, then smooth the seams so no cheese is exposed.
Season both sides with the salt and pepper and chill the stuffed patties for 10 minutes.
Toast the buttered buns in a heavy skillet over medium heat for 1 to 2 minutes, then remove them.
Cook the patties over medium heat for 5 to 6 minutes per side, turning gently once, until an instant-read thermometer inserted into the meat beside the cheese pocket reads 160°F.
Rest the patties for 3 minutes so the cheese settles and the beef finishes safely.
Build each burger with mustard, chopped onion, 3 pickle chips and one stuffed patty, then warn diners that the center is hot.
