Preheat the oven to 450ºF. Coat two baking sheets with cooking spray or olive oil.
Pat the broccoli dry, then combine it with the chicken, mozzarella, and pesto in a large bowl. Season with salt and pepper to taste.
On a lightly floured surface, roll out half of the dough into a 12 inch round. Mound half of the filling onto one half of the round, leaving a 1 inch border along the edges. Brush the edges with water. Fold the other half of the round over the filling and press or pinch the edges to seal. Repeat with the remaining dough and filling.
Transfer the calzones to the prepared baking sheet. Using a sharp knife, cut 5 slits diagonally across the top of each calzone. Brush oil over the top and sprinkle with coarse kosher salt. Bake until golden, about 15 minutes.
Meanwhile, heat marinara sauce in a medium saucepan over medium-low heat until hot, stirring occasionally. Serve alongside calzones.