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).
In a small bowl, combine all of the garlic-basil oil ingredients. Now is also a good time to gather and prep your fillings.
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.
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.)
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.)
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.
Once all of the turnovers have been filled and sealed, brush the surface with egg wash.
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.
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.
Store the pizza turnovers, well-wrapped, in the refrigerator for several days. Reheat, unwrapped, in a 350°F oven until warmed through.
