Prepare the dough. Combine all dough ingredients in a stand mixer or bread machine on the dough cycle. Then, knead until smooth or allow the bread machine to complete its cycle. If using a stand mixer, turn the dough out into an oiled bowl, cover it with plastic wrap or a damp cloth, and let it rise in a warm place for one hour or until doubled in bulk.
Prepare the baking sheet. Line a baking sheet with parchment paper. After that, place the dough on top and stretch it into a 10 x 10-inch square. Let it rest for ten minutes.
Preheat the oven. Preheat your oven to 375°F (190°C).
Shape the dough. Stretch the dough out to the full size of the baking sheet. Distribute the fillings evenly in this order: parmesan, mozzarella, pepperoni, leaving a 1-inch border on each side.
Roll and seal the dough. Carefully roll up the dough, pinching all around the loaf to seal in the filling. Flip the loaf over so the seal is on the bottom. Brush with the beaten egg and make slashes on top.
Bake. Bake for 40-50 minutes, or until the bread is golden brown. Allow it to cool for 15 minutes before slicing.