Set a pizza stone or baking sheet on an oven rack in the bottom third of the oven, and preheat to 425 degrees F.
Heat 1 1/2 tablespoons of the oil in a large nonstick skillet over medium-high heat. Add the ground beef, 3/4 teaspoon salt and half the pepper flakes, and cook, breaking the beef up into large chunks with a wooden spoon, until the beef is browned all over, about 3 minutes. Add the mushrooms, 2/3 of the garlic and 3/4 teaspoon salt, and cook for 3 minutes more. Fold in the spinach, and cook until wilted, about 1 minute (the mixture should be relatively dry). Transfer the mixture to a large bowl, and let cool completely.
Reduce the heat to medium. Add the remaining oil, garlic and pepper flakes to the skillet, and cook, stirring, to lightly brown the garlic, about 1 minute. Add the tomatoes and 3/4 teaspoon salt, and simmer the sauce until thick, about 6 minutes; set aside.
Divide the pizza dough into 12 balls. Spray 2 baking sheets with cooking spray. On a well-floured surface, roll out each ball into a 6- or 7-inch circle 1/8 inch thick. On the bottom half of each circle, mound 1/4 cup of the cooled beef mixture and top with 3 tablespoons of the mozzarella. Fold over the top half of the dough to form a semicircle, and pinch the edges to seal. Carefully transfer the calzones with a spatula to the prepared baking sheets. Lightly brush with the beaten egg. Cut 2 small slits on top of each to vent steam.
Refrigerate half the calzones. With a spatula, transfer the rest of the calzones to the hot pizza stone or baking sheet and bake until golden brown, about 22 minutes. Repeat with the remaining calzones. Reheat the tomato sauce, and serve on the side.
To freeze: Let the baked calzones cool completely, then wrap each one tightly in plastic wrap and then aluminum foil. Place them in freezer bags, and freeze for up to a month. The sauce can be frozen in 12 small freezer bags for up to a month.
To reheat: Thaw the frozen calzones in the fridge for 1 to 2 hours. Unwrap the calzones, then microwave them on a microwave-safe plate in 1-minute increments, rotating them after each, until heated through, 2 to 4 minutes. To reheat the sauce, microwave it in a microwave-safe bowl in 30-second increments, stirring after each, until the sauce is hot.
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