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Preheat the oven to 475°F with a rack in the center. Line a Crispiest Crust Pizza Pan with parchment. (See “tips,” below for other pan options.)
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On a lightly floured surface, use a rolling pin to roll the dough packet into a large square until it begins to resist (i.e., snaps back). Cover the dough and let it rest for 30 minutes. Meanwhile, make the garlic-basil oil (next step).
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In a small bowl, combine all of the garlic-basil oil ingredients. Now is also a good time to gather and prep your fillings.
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Roll the dough to a 15" square. As you roll, ensure the dough isn’t sticking to the surface by frequently lifting it up and flouring the surface underneath.
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Using a pizza wheel or sharp knife, trim about 1/4" off all sides to make a 14 1/2" square. Cut the square into four quadrants. (If your dough squares get soft or difficult to handle during the next part of the process, transfer them to the refrigerator, stacked on a plate or on a parchment-lined baking sheet, as you work.)
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Working with one square at a time, brush the edges with egg wash. On one half of the square, sprinkle 2 tablespoons (14g) grated mozzarella and 2 tablespoons (14g) fresh mozzarella in a triangular shape, leaving a 1/2" border around the edges. Dollop 4 teaspoons (19g) of the pizza sauce over the cheese, then sprinkle with 1 to 2 teaspoons Parmesan, to taste. Distribute 1/4 cup (15g to 40g) cooked vegetables or meat evenly over the sauce, then drizzle about 1 teaspoon garlic-basil oil over the filling. (Reserve the excess for finishing the turnovers.)
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Cut two slits in the bare side of the dough square (these will form the vents on top of the turnover), then fold it over the fillings diagonally to form a triangle. Use a fork to press and seal the edges, then transfer to the prepared pan. Repeat with the other three squares of dough and the remaining filling.
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Once all of the turnovers have been filled and sealed, brush the surface with egg wash.
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Bake for 12 to 14 minutes, until the turnovers are brown all over, puffed, and you can see layers in the crust along the edges.
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Remove the turnovers from the oven and immediately brush them with more garlic-basil oil and sprinkle with additional Parmesan. Let cool slightly on the baking sheet before transferring to a wire rack to cool for 10 to 15 minutes. Enjoy warm, with pizza sauce and garlic-basil oil for dipping, if desired.
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Store the pizza turnovers, well-wrapped, in the refrigerator for several days. Reheat, unwrapped, in a 350°F oven until warmed through.