Preheat the oven to 375ºF. Line a baking sheet with parchment paper.
Lightly flour your work surface then roll out the pizza dough to a rectangle measuring 11x13 inches.
In a small bowl mix together the shredded chicken with the Parmesan cheese and 1/2 cup of the marinara sauce. (Reserve the remaining 1/2 cup of marinara sauce for dipping.)
Position the dough so the longer side is closest to you then arrange the chicken mixture in the center of the dough. Top the chicken with the sliced mozzarella and then the basil leaves.
Beginning 1 inch from the top of the dough, cut 1/2-inch slits from the center of the dough outward toward the edges. Fold the top portion of the dough over then alternately braid the strips across the center. Cut off any excess dough then tuck the two end pieces underneath the stromboli. Transfer the stromboli to the baking sheet. (See Kelly's Notes.)
Brush the top of the stromboli with the olive oil, sprinkle it with additional Parmesan cheese then bake it for 25 to 30 minutes until it's golden brown and the cheese is melted. Remove the stromboli from the oven, transfer it to a rack and let it rest for 10 minutes. Slice and serve.