Line a baking sheet with a silicone baking mat or parchment paper sprayed with cooking spray.
Open up the pizza dough and pat out into an even 14-by-10-inch rectangle, with the longest side at the top. Cut into 32 rectangles by cutting 8 vertical strips, then cutting each strip 4 times horizontally.
Place a teaspoon of pizza sauce into the center of each rectangle, then top with a piece of mozzarella. Sprinkle on a few crushed red pepper flakes and top each with a slice of pepperoni. Place a hot dog across the shorter side (so it sticks out). Roll up each rectangle, pressing the seam to secure. Place seam-side down on the prepared baking sheet. Repeat with the remaining dough and fillings.
Beat the egg in a small bowl with 1 tablespoon water and brush over the top of the dough.
Mix together the Parmesan and Italian seasoning and sprinkle over the prepared pigs in blankets.
Bake until golden brown, about 15 minutes. Serve with the remaining pizza sauce.