Heat the oil in a heavy large skillet over medium-low heat. Add the garlic and saute until fragrant and tender, about 2 minutes. Add the tomatoes, thyme, parsley, and wine. Increase the heat to medium and simmer until most of the liquid evaporates, about 8 minutes. In a small bowl, season the shrimp with salt and pepper. Add the shrimp to the skillet and saute until just cooked through, about 2 minutes. Set aside to cool in a small bowl.
Position 1 oven rack in the center and the second rack on the bottom of the oven and preheat to 450 degrees F.
Line 2 heavy large baking sheets with parchment paper. Roll out each piece of pizza dough into a 7-inch-diameter floured round. Sprinkle half of the mozzarella over the lower half of each pizza dough, dividing equally and leaving a 1-inch border. Spoon the shrimp mixture over the cheese. Sprinkle the remaining cheese over the shrimp mixture. Brush the edge of the dough with egg wash. Fold the plain dough halves over the filling, forming half circles. Pinch the edges of dough firmly together and crimp to seal.
Transfer the calzones to the 2 prepared sheets. Bake until the calzones puff and become golden brown, about 12 to 15 minutes. Brush extra-virgin olive oil over calzones and serve.
Mix the warm water and yeast in a small bowl to blend. Let stand until the yeast dissolves, about 5 minutes. Mix the flour and salt in a food processor to blend. Blend in the oil. With the machine running, add the yeast mixture and blend just until the dough forms. Turn the dough out onto lightly floured surface and knead until smooth, about 1 minute. Transfer the dough to a large oiled bowl and turn the dough to coat with the oil. Cover the bowl with plastic wrap and set aside in a warm draft-free area until the dough doubles in volume, about 1 hour. Punch the down dough and form into ball. The dough can be used immediately or stored airtight in the refrigerator for 1 day.