Prepare the oven: Heat the oven to 375°F. Line a large baking sheet with parchment paper.
Roll the dough: Gently roll each dough sheet to about 12 by 8 inches. If using crescent rolls, pinch all perforations closed first.
Cut the squares: From each sheet, reserve a 12-by-2-inch strip. Cut each remaining 12-by-6-inch rectangle into eight 3-inch squares. You will have sixteen squares total.
Cut the bandages: Cut each reserved dough piece into eight 12-inch-long strips about 1/4 inch wide. Cut each strip into three 4-inch pieces, making forty-eight bandages total.
Fill the pockets: Arrange eight dough squares on the baking sheet, about 1 1/2 inches apart. Add 1 teaspoon sauce, 2 tablespoons shredded mozzarella, and the chopped equivalent of 3 pepperoni slices to each. Leave a clean 1/2-inch border.
Seal: Cover each with a dough square. Gently press out large air pockets, pinch all edges together, and crimp with a fork.
Decorate: Beat the egg with the water. Brush a little over the lids. Arrange six bandages across each pocket, leaving a small eye opening near the top. Trim excess length and press the strip ends onto the pocket edges.
Vent and brush: Make one small vent through each lid in the eye opening. Do not cut through the base. Brush the bandages lightly with egg wash.
Bake: Bake for approximately 16 to 20 minutes, until golden on top, lightly browned underneath, and cooked through at the seams. If needed, loosely cover with foil and continue baking until the dough is done.
Add eyes: Rest for about 5 minutes. Add two small mozzarella eye bases and two olive pupils to each pocket. Serve warm with extra pizza sauce.