Preheat oven to 375 degrees in a large oven-safe skillet, heat butter and 1 tablespoons of the oil on medium-high. Once butter foams, add mushrooms and a pinch of salt and cook, tossing occasionally, for 5 minutes.Add shallot and cook, tossing occasionally, until mushrooms are golden brown, 2 to 3 minutes; transfer to plate and wipe the skillet clean.
Return the skillet to medium heat. Rub chicken with remaining 1 tablespoon oil and season with 1/4 teaspoon salt and pepper. Add chicken to skillet, skin side down and cook until browned, 10 to 12 minutes; drain any excess fat. Turn chicken over; add wine, broth, and thyme sprigs. Transfer the skillet to the oven until chicken is cooked through (165 degrees) 5 to 6 minutes.
Transfer chicken to a plate, discard thyme, and return the skillet to medium heat. Stir in cream and mushroom mixture and cook until heated through, about 2 minutes. Serve with chicken. Garnish with thyme, if desired.
Christopher Lee