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Divide the dough into four 12-ounce pieces and shape into balls. Transfer balls to a lightly greased 9- by 13-inch baking dish and brush with oil. Cover tightly with plastic wrap and refrigerate for at least 36 hours or up to 48 hours.
Place a pizza stone under the oven broiler; heat broiler and let pizza stone heat for 30 minutes.
For the pizza: Heat the olive oil in a large skillet over medium-high heat. Add the sausage and cook, stirring and breaking up the pieces with a wooden spoon, until browned but not completely cooked through, about 4 minutes. Add the wine and cook, scraping the bottom of the skillet, until reduced, about 2 minutes. Set aside and let cool slightly.
Working with one dough ball at a time, dust heavily with semolina. Using your fingertips, press the dough from the center out into a 10-inch circle, about 1/4-inch thick and leaving a 1-inch-thick crust around the edges. Hold the dough straight up, and with fingertips circling crust, let dough naturally fall. Slide fingers around crust in a circular motion as you would turn a steering wheel until the dough in the center is stretched to about 1/8-inch thick. Transfer to a semolina-dusted pizza peel.
Spread about 1/2 cup of the pistachio pesto evenly over the center of the pizza. Evenly distribute one-quarter of the sausage mixture, mozzarella and basil leaves on top. Sprinkle with 1/4 ounce of the Pecorino and drizzle with oil. Slide the pizza onto the stone and bake until the cheese melts and the crust begins to puff and char in spots, 3 to 4 minutes. Transfer the pizza to a plate using the pizza peel. Repeat with the remaining 3 pizzas.