Heat a large skillet over medium-high heat, then add 2 tablespoons of olive oil. Add onions and garlic and cook for 2 minutes.
Add the chicken and cook for 7 minutes. The chicken should no longer be pink.
Add onions and garlic and cook for 2 minutes.
Add broccoli and carrots, then cook for about three minutes.
Add peas and cook 2 minutes.
Stir in the cooked pasta and the remaining tablespoon of oil. Pour in the chicken broth, vinegar, thyme, salt, and pepper. Cook for one minute then stir in Italian parsley.
Transfer to a serving platter and sprinkle with Parmesan cheese if desired.