Preheat oven to 375 degrees.
Bring a pan sprayed with nonstick spray to medium heat on the stove. Add veggies and cook for about 3 - 4 minutes, until they have softened slightly. Remove from heat. Once cool enough to handle, blot away as much excess moisture as possible with paper towels.
Carefully skewer a mushroom onto each stick, and slide it down to about 2 inches from the end. Add a few pepper and onion chunks to each stick, and then add another mushroom. Skewer the rest of the pepper and onion chunks, and then add the final mushroom to each. If needed, adjust veggies so that the final mushroom covers the tip of the stick.
Roll out the pizza dough and cut it into two squares, each at least 6 inches by 6 inches. Distribute the cheese strings vertically over the dough squares, leaving 1-inch borders on all sides.
Lay one veggie-stacked stick along the left edge of one cheese-covered dough section, with the top of the stick about an inch down from the top of the dough. Repeat with the other veggie-stacked stick.
Beginning with one dough square, bring the left edge up over the veggies and roll over to the right, until the dough is completely wrapped around the veggies -- pinch the top, bottom, and sides of the dough to close and seal, and squeeze the dough gently around the veggies to make sure it holds on to the fillings. Repeat with the other square of dough.
Gently place the two loaded sticks on a baking sheet sprayed with nonstick spray. Bake in the oven for 13 - 15 minutes, until dough is golden brown and crispy. Remove from the oven and carefully (skewers will be hot) place on paper towels to cool for 3 - 5 minutes. Serve with tomato sauce for dipping!