Special equipment: A baking stone or baking sheet
Position an oven rack in the bottom position of the oven, put a baking stone or a baking sheet on it and preheat the oven to 375 degrees F. Line another baking sheet with parchment.
Toss and squeeze together the mozzarella, salami, pepperoni, Parmesan, 4 tablespoons of the breadcrumbs, 1 of the eggs, banana peppers, oregano, 1/4 teaspoon salt and red pepper flakes in a medium bowl; set aside.
Dust a work surface with flour, and unfold the puff pastry sheet onto it. Roll it out into a 9-by-15-inch rectangle. Transfer the pastry to the prepared baking sheet.
Sprinkle the remaining 1 tablespoon breadcrumbs lengthwise down the center of the pastry. Pile and pack the cheese mixture into a log on top of the breadcrumbs, leaving a 1-inch border on the 2 short ends and about a 3-inch border on the 2 long sides. Fold the 1-inch borders to overlap the filling. Use kitchen shears to snip inch-wide strips of dough along both long sides that stop 1/2 inch from the filling, like a kind of fringe.
Brush the pastry all over with some of the remaining beaten egg. Starting at 1 short end, alternate folding the strips, on a slight diagonal, over the filling to make a design that looks vaguely like a braid. The filling should be mostly covered, but some will peek out from the spaces between the strips. Holding both ends of the roll, bend it into a U shape. Brush the top and sides with more egg. Sprinkle oregano, red pepper flakes and Parmesan all over the top.
Put the baking sheet with the pizza roll on top of the preheated baking sheet, and bake until golden brown, 40 to 45 minutes.
Meanwhile, heat the pizza sauce. Use a serrated knife to slice the roll into pieces. Serve the pizza sauce on the side for dunking.
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