Cut chicken breasts in half horizontally, so you have four thinner pieces of chicken. Place plastic wrap over pieces of chicken and pound out the chicken breast with a meat tenderizer until they are equal thickness.
On a plate, combine flour, salt and pepper. Coat chicken pieces well in flour mixture on all sides.
In a large skillet over medium high heat, add 1-2 tablespoons avocado oil. Add chicken and cook 4-5 minutes on each side, or until both sides are golden. Transfer chicken to a plate and set aside.
To the same pan, add additional oil, if necessary, and then add onion, garlic and fresh thyme leaves. Sauté aromatics for 3-4 minutes, or until fragrant. Add sliced baby portobello mushrooms and cook for another 5-6 minutes.
Add chicken broth, marsala wine and balsamic vinegar. Bring the sauce to a boil, then reduce to a simmer, cover and simmer for 10 minutes, allowing sauce to reduce and thicken.
Add chicken back into sauce along with half and half or dairy free creamer. Simmer everything for another 3-4 minutes. Serve with your favorite side, such as rice, pasta or mashed potatoes, and garnish with fresh parsley.