Boil pasta according to package directions in salted water in a very large skillet or pot. In the final 2 to 3 minutes of cooking, add the broccoli and cauliflower florets. Drain but reserve 1/2 cup pasta water in case you need it in step 6.
Return the pasta and vegetables to the cooking pot; set aside.
To a large skillet, add the olive oil, chicken, and cook for about 3 minutes.
Add the carrots, onions, salt, pepper, and cook for about 5 minutes, or until chicken is cooked through and vegetables have softened; stir and flip intermittently to ensure even cooking.
Add the garlic and cook for about 1 minute, or until fragrant; stir nearly constantly.
Add the chicken and carrot mixture to the large pot with the cooked pasta, broccoli, and cauliflower.
Add the spinach, cheeses, and stir until cheese has all melted and spinach has wilted. Taste and check for seasoning balance and if necessary, add additional salt and/or pepper. Serve immediately.
