Set a pizza stone or baking sheet on an oven rack in the bottom-third position and preheat to 425 degrees F.
Heat 1 teaspoon of the olive oil in a large nonstick skillet over medium-high heat. Add the ground beef, 1/4 teaspoon salt, and a pinch of crushed red pepper. Break the beef into large chunks with a wooden spoon. When the beef is browned all over, about 2 minutes, add the mushrooms, 2/3 of the garlic and 1/4 teaspoon salt. Cook for 3 minutes, then fold in the spinach to wilt, about 1 minute. This mixture should be relatively dry. Remove to a large bowl to cool completely.
Reduce the heat to medium. Add the remaining 1 teaspoon olive oil and garlic and another pinch of crushed red pepper to the skillet. Stir for a minute to lightly brown the garlic, then add the tomatoes and 1/4 teaspoon salt. Simmer until thick, about 6 minutes. Reheat this sauce when ready to serve with the calzones.
Divide the pizza dough into 4 balls. Spray a baking sheet with cooking spray. On a well-floured surface, roll out each rested ball into a 6- or 7-inch circle 1/8 inch thick. On the bottom half of each circle, mound a quarter of the cooled beef mixture, 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 sheet. Lightly brush with the beaten egg. Cut 2 small slits on top of each to vent steam.
Bake the calzones in the oven on the hot pizza stone or baking sheet until golden brown, about 22 minutes. Serve with the heated tomato sauce on the side.
Stir together the warm water, honey and yeast in a measuring cup or small bowl. Let sit until a small layer of foam develops at the top, 3 to 5 minutes. (If this doesn't happen, discard and try again with new yeast.)
Whisk the flour and 1/2 teaspoon salt together in a medium bowl. Add the foamy yeast mixture and olive oil and mix together with a stiff rubber spatula. Form the dough into a uniform ball in the center of the bowl. Be sure to scrape and use any dough stuck to the sides. The dough will be very sticky at this stage. Cover the bowl tightly with plastic wrap and keep in a warm place until the dough has doubled in size, about 2 hours.
Lightly dust a baking sheet with flour. Sprinkle a generous amount of flour on a clean, dry work surface. Scrape the dough onto the floured surface and knead for a couple of minutes, working in just enough flour to make the dough less sticky but still moist to the touch. Form the dough into 1, 2 or 4 balls, for large, medium and small pizzas. Place the dough ball(s) on the prepared baking sheet. Cover loosely with plastic wrap or a dish towel and let rest for 30 minutes. The dough can then be shaped and cooked as desired or wrapped well and frozen for up to 1 month. Thaw frozen dough at room temperature, shape and cook as desired.
We love using white whole-wheat flour. It's lighter in color and milder in flavor than its regular whole-wheat cousin (a red wheat variety), which can be denser and harsher. Both flours are almost the same, nutritionally; our pizza dough has about 9 grams of fiber per serving. Be sure to check out our Cook's Note, below, a tip on making quick pizza for rushed weeknight dinners.
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