Preheat the oven to 375 degrees F and line a baking sheet with parchment.
Combine the tomato sauce, basil, oregano, garlic powder and 1/2 teaspoon salt in a microwave-safe bowl and stir until smooth. Microwave the sauce, stirring halfway through, until hot, 1 1/2 to 2 minutes. Let cool to room temperature.
Put the pizza dough on a lightly floured work surface and roll into a 10-by-14-inch rectangle. Spread 3 tablespoons of the pizza sauce evenly over the dough and sprinkle with the mozzarella and pepperoni pieces.
Working from one long side, roll up the rectangle like a jelly roll and position it seam-side down on a cutting board. Use a paring knife to trim the ends of the dough log, then cut the log in half lengthwise. Lay the 2 halves cut-side up next to each other and twist them together into a tight loaf about 12 inches long.
Transfer the twisted dough to the prepared baking sheet and brush gently all over with the egg mixture. Bake until the bread is golden brown and the cheese is melted in the center, 30 to 32 minutes. Transfer the braid to a cutting board, cut into slices and serve with the remaining pizza sauce for dipping.
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