Preheat oven to 425°.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add artichoke hearts, mushrooms, and garlic; cook 6 minutes or until liquid is absorbed. Remove from heat.
Combine artichoke mixture, cheeses, basil, and pepper in a medium bowl. Set aside.
Divide pizza dough evenly into 4 pieces on a lightly floured surface; roll each piece into a round disk. Let stand 5 minutes. Pat or roll each disk into a 5-inch circle.
Arrange 1 prosciutto slice over each circle. Spoon 1/2 cup cheese filling on half of each circle, leaving a 1/2-inch border. Brush edges with water. Fold pizza dough over the cheese filling until edges almost meet. Bring bottom edge over top edge; crimp edges of dough with fingers to form a rim.
Place calzones on a baking sheet coated with cooking spray. Coat calzones with cooking spray. Cut 3 small slits into top of dough using a sharp knife.
Bake calzones at 425° for 17 minutes or until lightly browned. Serve with pasta sauce.