Combine the ground beef, mozzarella cheese, bread crumbs, barbecue sauce, and pepper. Shape into 4 burgers (or 8 smaller slider-size burgers).
Heat a grill pan over medium-high heat on the stovetop or prepare an outdoor gas or charcoal grill for direct medium-high heat.
Grill for about 5 minutes on each side, or until the burgers are cooked through and at least 160 F in the center (well done—USDA minimum safe temperature). If you prefer medium, cook the burgers to 145 F.
Brush with more barbecue sauce at the last minute or two, if desired.
Place a piece of mozzarella cheese on the burgers and continue heating until the cheese has melted.
Toast the buns, if desired, and fill with the burgers and toppings.