1. Preheat oven to 450°. On a lightly floured surface, divide dough into four equal pieces. Using a lightly floured rolling pin or your hands, roll or stretch each piece into a 6-inch round. If dough shrinks, let it rest about 5 minutes, then roll again.
2. Divide mozzarella among the dough rounds, then top equally with pesto, broccoli, and chicken.
3. Gently pull half of dough over filling to make a half-moon shape. Fold bottom edge over top edge and pinch firmly to seal. Brush calzone tops lightly with olive oil. Transfer to a large baking sheet.
4. Bake calzones on middle rack until golden brown, 20 to 25 minutes (15 to 20 minutes in a convection oven). Let cool about 5 minutes, then transfer to plates.
Note: To make in advance, prepare through step 3, then freeze on a baking sheet. When frozen, wrap each calzone with aluminum foil and keep in freezer for up to 1 month. Bake at 375° for 30 to 35 minutes, until browned and cooked through.
Note: Nutritional analysis is per serving.