If you have a pizza stone, place it in the oven. In a bowl, combine the cheeses, prosciutto and basil. Season with salt and pepper.
On a floured surface, roll out each ball of dough to 1/4-inch thickness, and form circles 8 to 10 inches in diameter. Place one-quarter of the filling on one side of each circle. Moisten the edges of the dough with water, then fold dough over the filling. Seal by crimping the edges with a fork. Brush the tops and edges of the calzones with the egg mixture. Trim the excess dough with a pizza cutter, and make a small incision on the top to allow steam to release.
Place calzones on the pizza stone or parchment-lined baking sheet, and bake until golden, 18 to 20 minutes. Serves 4
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