Preheat the oven to 375°F (190°C).
In a large skillet, heat olive oil over medium heat. Add the onions and garlic, sautéing until translucent.
Toss in the red and yellow bell peppers, zucchini, yellow squash, broccoli, and carrots. Season with salt, pepper, and Italian seasoning. Cook until the vegetables are just tender.
In a large mixing bowl, combine the shredded chicken, cooked vegetables, and cooked penne pasta. Stir in the heavy cream and half of the mozzarella cheese until well mixed.
Transfer the mixture to a greased 9x13 inch casserole dish. Sprinkle the remaining mozzarella cheese and Parmesan cheese on top.
Bake in the preheated oven for 25-30 minutes, or until the cheese is bubbly and golden brown.
Remove from the oven and let it rest for a few minutes before serving.
