1. In a frying pan, heat 1 tablespoon of butter and 1 tablespoon olive oil on medium heat until the butter melts. Add the mushrooms, carrots and shallots. Cook, stirring occasionally for 5 minutes. Remove the vegetables from the pan.
2. Add the rest of the butter and olive oil and heat until the butter melts. Salt and pepper the chicken on all sides and than add the pieces to the pan, browning on all sides for about 5 minutes.
3. Add the vegetables, vinegar and tarragon. Cover and simmer on low-medium heat for 20 minutes or until the chicken is cooked. Remove chicken and vegetables to a serving dish.
4. Continue cooking the vinegar on high heat until reduced to about a third (about 8 minutes). Stir in cream and pour sauce over chicken and vegetables.

Makes 6 - 8 servings