Preheat oven to 400 degrees Fahrenheit (200 degrees Celsius). Soak 12 wooden 12-inch skewers in cold water for 30 minutes.

Place all the vegetables except for the cherry tomatoes in a large bowl and drizzle with three tablespoons olive oil. Season generously with salt, black pepper, garlic powder, and Italian seasoning. Toss until well coated. Arrange on a baking sheet in a single layer. Roast for 20 minutes. Remove from oven and set aside.

While the vegetables are roasting, roll out the pizza dough into a 24- by eight-inch (60- by 20-centimeter) rectangle.

Using a sharp knife, slice into 12 two- by eight-inch (five- by 20-centimeter) strips.

Spread one teaspoon of pesto down the center of each strip and sprinkle with one tablespoon of shredded mozzarella.

Fold each strip in half vertically and press edges together to seal.

When the vegetables are cool enough to handle, thread four mushrooms onto four of the skewers, about one inch (two and a half centimeters) apart, leaving a small space at the top. Repeat with four cherry tomatoes on four additional skewers and a combination of peppers and onions on the remaining four skewers.

Starting at the pointed end of the skewer, weave a dough strip down and around the vegetables, pinching at the bottom to secure, and place on a baking sheet lined with Gefen Parchment-. Repeat with the remaining skewers, placing six skewers on each sheet.

Brush the skewers with olive oil and season with additional salt, pepper, garlic powder, and Italian seasoning. Sprinkle lightly with Parmesan cheese. Bake for 12–15 minutes until lightly browned, switching the pans between upper and lower racks midway through baking. Serve immediately.