Remove pizza dough from refrigerator; let stand at room temperature 1 hour.
Prepare grill to medium-high heat.
Coat onion slices and bell pepper pieces with cooking spray. Place vegetables and sausages on a grill rack coated with cooking spray. Grill vegetables 4 minutes on each side or until browned; grill sausages 8 minutes or until done, turning occasionally to brown on all sides. Remove vegetables and sausages from grill; cool slightly. Cut onion slices in half; cut bell pepper pieces into 1/2-inch strips. Cut sausages diagonally into thin slices.
Preheat oven to 500°
Place dough on a lightly floured surface; divide dough into 4 equal portions. Roll each portion into a 9 x 5-inch rectangle. Spread 1/4 cup pizza sauce evenly over each rectangle, leaving a 1/4-inch border. Arrange sausage evenly over half of each rectangle; top evenly with onion and bell peppers.
Sprinkle each calzone with 1/3 cup cheese. Fold other half of dough over filling; press edges together with a fork to seal.
Place calzones on a baking sheet coated with cooking spray. Coat calzones with cooking spray. Bake at 500° for 15 minutes or until golden brown. Remove from oven; let stand 5 minutes.
Cut each calzone in half; serve with remaining New York-Style Pizza Sauce.