Preheat oven to 325 degrees.
Add 1 TBS oil to an oven safe sauté pan or dutch oven on medium high heat.
Sprinkle chicken breasts with 1 tsp salt and 1/2 tsp pepper. Add to pan and sear on both sides until browned about 4 minutes (it will finish cooking in the oven).
Remove chicken to a plate. Add the butter to the sauté pan.
Once the butter is melted, add the mushrooms and let brown, 4-5 minuets on each side.
Add shallots and garlic and sauté for two minutes.
Meanwhile in a separate bowl, mix the coconut milk and the balsamic vinegar together. Set aside.
Slowly pour in the wine or stock to the sauté pan, scrapping up the browned bits from the bottom of the pan to deglaze.
Sprinkle in 1 tsp salt and 1/4 tsp pepper.
Add the chicken back to the pan. Pour the coconut mixture over the chicken.
Cover with a lid and bake for 20 minutes or until an instant read thermometer reads 165 degrees when inserted into the middle of the thickest part of the chicken.
Remove from oven. Taste sauce and add any additional salt or pepper if desired.

If your sauce needs thickened, return the pan to the stove over medium high heat. Add the arrow root and water mixture to the sauce and stir for about three minutes.
Garnish with parsley and serve over basmati rice or with roasted vegetables or savory sweet mashed potatoes.