Preheat the oven to 450˚. Line a baking sheet with parchment paper. Combine the chicken, mozzarella, spinach, roasted red peppers, 1/2 teaspoon salt and a few grinds of pepper in a large bowl.
Divide the pizza dough into 4 pieces; keep covered with plastic wrap until ready to use. Working with 1 piece at a time, roll the dough into a ball and dust with flour, then roll into a 7-inch round on a lightly floured surface. Spoon one-quarter of the chicken mixture on one side of the dough round, then fold the dough over the filling and crimp the edges with a fork to seal. Transfer to the prepared baking sheet. Repeat with the remaining dough and filling.
Brush the calzones with the beaten egg and make a few slits in the top of each with a paring knife. Bake until the calzones are golden and puffed, about 25 minutes. Let rest 5 minutes. Serve with marinara sauce.