Heat a griddle to medium-high heat.
Combine the basil, shallot, and mayonnaise, and reserve.
Take each meatball and press one cube of cheese into the center of each, wrapping the meat around to completely seal.
Place each meatball on the griddle and flatten to about 2 inches high and cook about 4 minutes per side, or until the cheese is fully melted.
Place the meatballs on a plate and allow to rest.
Wipe the griddle mostly clean and toast the slider rolls on the griddle.
To assemble: Dress the bottom of each roll with some of the mayonnaise. Top with one of the meatballs, some of the pickled peppers, then dress the tops of each roll with a bit more of the mayonnaise.