In a large saucepan, heat the olive oil to medium-high heat.
Butterfly the chicken breasts or you can pound them to thin them out. This is so it cooks evenly.
Put 3 tablespoon of arrowroot powder on a plate and light coat each chicken breast in. Be sure this is not a heavy coating, but the entire breast should be covered. Check to make sure your oil is hot- it should sizzle when you put a drop of water in it.
Put the chicken in the pan and cook on one side for 3 minutes, then turn and cook on the other side for about 3 minutes. Remove from pan and set aside.
Add the shallots, mushrooms, and garlic to the pan and saute for about 5 minutes. Then add chicken broth, coconut milk, balsamic vinegar, thyme, salt, and pepper to the pan. Whisk the sauce together then add ½ teaspoon of arrowroot powder, set the rest aside, and add later if needed. Whisk the arrowroot powder in thoroughly.
Let the sauce simmer for 10 minutes. Check at 5 minutes and if it has not thickened at all, whisk in another ½ teaspoon of arrowroot powder.
Add chicken back to the sauce. Cover the pan and saute for 5-10 minutes or until chicken is cooked all the way through and sauce is the consistency of gravy.
Top with fresh parsley and enjoy!