Heat the oven to 400 F. Lightly coat a baking sheet with cooking spray.
In a medium bowl, add the asparagus, spinach, broccoli, mushrooms and garlic. Drizzle 1 teaspoon of the olive oil over the vegetables and toss to mix well.
Heat a large, nonstick frying pan over medium-high heat. Add the vegetables and saute for 4 to 5 minutes, stirring frequently. Remove from heat and set aside to cool.
On a floured surface, cut the bread dough into quarters. Press each piece into a circle. Using a rolling pin, roll the dough into a thin oval. On half of the oval, add 1/4 of the sauteed vegetables, 1/4 of the tomato slices and 2 tablespoons cheese. Wet your finger and rub the edge of the dough that has the filling on it. Fold the dough over the filling, pressing the edges together. Roll the edges and then press them down with a fork. Place the calzone on the prepared baking sheet. Repeat to make the other calzones.
Brush the calzones with the remaining 1 teaspoon olive oil. Bake until golden brown, about 10 minutes.
Heat the pizza sauce in the microwave or on the stove top. Place each calzone on a plate. Serve with 2 1/2 tablespoons pizza sauce on the side or pour the sauce over the calzones.