Preheat the oven to 350 degrees F. Coat a 9x13 baking dish with cooking spray and set aside. Line a baking sheet with parchment and spray lightly with cooking spray.
Place the chicken in the prepared baking dish and bake in the preheated oven for 20-25 minutes, or until juices run clear and meat is cooked through. Let chicken cool and shred. Increase oven temperature to 400 degrees F.
Lightly flour a work surface. Roll dough into a 14x10-inch rectangle. Spread with marinara sauce, leaving a 1-inch border around the dough edges on all sides. Top with the shredded chicken, mozzarella cheese and Parmesan cheese.
Fold 1 inch of dough over the filling on the short sides of the rectangle. Fold the long ends over the filling, pinching the dough together at the corners. Roll the dough, starting at one of the long ends, as tightly as possible. Pinch the seam to seal.
Transfer the roll, seam side down, to the prepared baking sheet and brush with the egg. Bake in the preheated oven for 25 minutes, or until the bread is golden brown.
Loosen the stromboli from the parchment using a spatula. Transfer to a cutting board, cut into 1-inch slices and serve immediately!