Preheat oven to 425°. Heat 1 Tbsp. olive oil in a small saucepan over medium heat. Add garlic and cook until fragrant, about 30 seconds. Stir in tomato sauce, Italian seasoning, ½ tsp. salt and ¼ tsp. pepper. Simmer for 15 minutes. Remove from heat. Stir in basil. Set aside.
Heat remaining 1 Tbsp. olive oil in a large skillet over medium high. Add ground chuck and cook, breaking into pieces, for 6 minutes, or until browned. Drain. Return meat to the pan. Add mushrooms, onions, pepperoni, and remaining 3/4 tsp. salt and ¼ tsp. pepper. Cook, stirring occasionally, until mushrooms are tender, about 6 minutes. Remove from heat and stir in ¾ cup tomato mixture.
Divide dough into 4 pieces, 4 oz. each. Roll each into a 7-inch circle. Spoon ¼ meat mixture (about ½ cup each) and ¼ cup mozzarella on to half of a circle. Moisten edges with lightly beaten egg, fold dough over and crimp to seal edges. Cut 3 small slits in the top. Repeat with remaining ingredients.
Transfer calzones to a parchment-lined baking sheet and lightly brush with egg. Bake at 425° for 20 minutes or until golden brown. Serve with remaining tomato mixture.