Fill a stock pot or dutch oven with water, heat to boiling, and plunge the sliced mushrooms into pot for 1 - 2 minutes, remove and place into a colander. Pat dry with paper towel and set aside.
Heat the olive oil and butter in a skillet or heavy bottomed panned over medium-high heat and cook the chicken until the outside is a golden brown, turn, and cook other side until golden brown approximately 15 minutes. Remove from heat. Place chicken on plate and cover with foil to keep warm.
Place shallots in skillet with remaining butter/oil mixture. Add mushrooms and cook until golden brown. Remove from heat, pour in rum, return to heat and boil for 1 minute. Add crème fraiche and taste for seasoning - add more salt and pepper if necessary. Boil for 2-4 minutes, then serve immediately over chicken breasts and garnish with parsley.
Recipe by Tout va bien! (tm) at https://christymajors.com/the-french-winophiles-chicken-chinon/