Let the dough warm and rise in the morning. Liberally sprinkle flour on a cutting board. Place unwrapped pizza dough on the board, turning a few times to shape into a thick rectangle and cover lightly with flour. Cover with plastic wrap to prevent drying and let sit for at least 6 hours to fully rise so the result is light and fluffy, not dense.
To make the dressing, whisk together in a small bowl the oil, 3 tablespoons marinara sauce, garlic, lemon juice, oregano, salt, and pepper.
Leaving several inches at the bottom, thread the skewers in an alternating fashion with peppers, onion, mushrooms, and sausage, repeating until you’ve used up all the ingredients. Arrange the kebabs on a large platter. Use a brush or spoon to cover each kebab with dressing. Reserve a couple of tablespoons of the dressing.
Lightly oil the grill and preheat on medium heat. Remove the plastic wrap from the dough. Sprinkle with more flour to keep from being sticky. Press the dough into a larger rectangle, roughly 6 inches long. Estimate the width of the rectangle by the number of kebabs you are making as you’ll need one strip that’s about 1 ½ inches for each skewer. Use a sharp knife to cut strips lengthwise. Starting at the sharp end of each skewer, pierce the end of the dough strip and wrap it in a spiral around the skewer, allowing veggies and sausage to poke through. Wrap the end of dough around the bottom of the skewer and pinch to fasten. Arrange on the platter. Brush remaining dressing over the dough. Place skewers on the hot grill and cook for about 5 to 7 minutes. Use a spatula if necessary to help gently flip the kebabs. Cook for another 5 minutes, watching to ensure the dough doesn’t burn. Turn the heat down if the dough looks like it’s starting to burn. In the last couple of minutes, sprinkle shredded cheese on top. Warm the remaining marinara sauce, if serving. Remove kebabs from the grill and serve with marinara sauce on the side.