Preheat oven to 350 degrees F (175 degrees C). Grease a baking sheet.
Divide pizza dough into 4 equal parts. Roll each piece into a large oval on a lightly floured surface.
Mix spinach, mushrooms, zucchini, cabbage, green bell pepper, fresh tomato, onion, sun-dried tomatoes, artichoke hearts, olives, garlic, and Italian seasoning in a large bowl. Season with salt and black pepper. Divide vegetable mixture over the center of each piece of dough, leaving about a 1-inch border of dough. Sprinkle about 1/4 cup mozzarella cheese and about 2 tablespoons Asiago cheese over vegetables
Fold pizza dough in half to create a half-moon; pinch and seal the edges of each calzone. Transfer calzones to the prepared baking sheet. Cut small holes on the top of each to vent; brush tops with olive oil.
Bake in the preheated oven until lightly browned, about 30 minutes. Sprinkle remaining mozzarella cheese and Asiago cheese evenly over each calzone and continue baking until cheese is melted and crust is golden brown, 5 to 10 minutes more. Serve with marinara sauce for dipping.