For the dough: Lightly oil a large bowl. Combine the yeast, salt, sugar and 2 1/4 cups of the flour in the bowl of a stand mixer fitted with the paddle attachment and stir to combine. With the mixer running, add the water and oil. Beat until the dough comes together, about 2 minutes. If the dough is too wet, add more flour, 1 tablespoon at a time, until it is still soft and slightly sticky, but you can turn it out. Transfer the dough to the oiled bowl and cover with plastic wrap. Let stand until doubled in size, about 1 hour.
For the calzones: Preheat the oven to 500 degrees F. Line a large rimmed baking sheet with parchment paper.
Mix together the mozzarella, ricotta and Parmesan in a medium bowl. Set aside.
Lightly flour a work surface. Turn out the dough and divide it into 4 pieces. Gently pull and stretch each piece of dough into a 7- to 8-inch round, trying to keep the thickness of the dough fairly even. Lay out the rounds and spoon one-quarter of the cheese mixture onto one side of each round (about 1/2 cup per round). Spoon about 2 tablespoons of the Homemade Pizza Sauce onto the cheese. Add up to 1/3 cup of your preferred topping onto each of the calzones, then fold the side of the dough over the fillings. Fold and crimp the edges to seal them completely.
Transfer the calzones to the prepared baking sheet and brush them lightly with oil. Use a sharp paring knife to cut 1 or 2 vents into each, then bake until deep golden brown, about 18 minutes.
Serve with additional Homemade Pizza Sauce on the side, if desired.
Homemade Pizza Sauce:
Yield: 2 1/2 cups (about 6 calzones)
Heat the oil in a medium saucepan over medium-high heat. Add the oregano, garlic and red pepper flakes and cook, stirring, until the garlic is golden brown, 1 to 2 minutes. Add the tomatoes, basil and 1/2 teaspoon salt and bring to a simmer. Reduce the heat to medium and simmer, with the saucepan loosely covered in foil to avoid splattering and stirring occasionally, until fairly thick, 5 to 10 minutes, depending on your tomatoes. Remove the basil.