Pulse 5 oz of the pepperoni and the garlic cloves in the bowl of a food processor until finely minced. Set the remaining pepperoni aside.
In a bowl, combine the ground beef, minced pepperoni, garlic, salt, pepper, and basil. Toss to mix and gently form into patties large enough for the slider rolls. Press your thumb into the center of each to make a small indentation. Cover and let rest in fridge for 30 minutes to an hour.
Meanwhile, clean your grill grate and preheat it to medium high.
Place patties on grill and cook 5 to 7 minutes, flipping once. Arrange buns, cut side facing up on cool side of grill to toast. Place a few slices of pepperoni on each and top with mozzarella slices. As it melts it will hide the pepperoni and melt up towards the tops of the buns.
Serve the patties with a fresh basil leaf and a spoonful of warmed marinara sauce topped with the cheese bun.