Put a baking sheet in the oven and preheat to 450 degrees F. On a lightly floured surface, divide the dough into quarters; cover with a towel.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the pork, 3/4 teaspoon salt and 1/2 teaspoon pepper and cook, breaking up the meat with a wooden spoon, until no longer pink, 3 to 4 minutes. Add 2 teaspoons paprika and cook, stirring, 30 seconds. Stir in the bell pepper and cook until slightly softened, about 2 more minutes. Transfer to a bowl and stir once or twice to cool slightly.
Pat and stretch each portion of dough into a rough 8-inch round. Pile the pork mixture and mozzarella on one side of the rounds, leaving a 1/2-inch border, then fold the dough over the filling. Pinch and crimp the edges to seal. Mix 1/2 tablespoon olive oil with the remaining 1/2 teaspoon paprika; brush over the calzones. Cut a 1-inch vent in each.
Remove the baking sheet from the oven and add the calzones. Bake until golden brown, 12 to 15 minutes. Meanwhile, toss the greens with the vinegar and the remaining 3 tablespoons olive oil in a bowl and season with salt and pepper. Serve with the calzones.