Heat oven to 375°F. In large oven-safe skillet on medium-high, heat butter and 2 tablespoons oil. Once butter foams, add mushrooms and pinch salt and cook, tossing occasionally, 5 minutes. Add shallot and cook, tossing occasionally, until mushrooms are golden brown, 2 to 3 minutes; transfer to plate and wipe skillet clean.

Return skillet to medium. Rub chicken with remaining 1 tablespoon oil and season with 1/4 teaspoon each salt and pepper. Add chicken to skillet, skin side down, and cook until browned, 10 to 12 minutes; drain excess fat. Turn chicken over; add wine, then broth and thyme. Transfer skillet to oven and bake until chicken is cooked through (165°F), 5 to 6 minutes.

Transfer chicken to plate, discard thyme and return skillet to medium heat. Whisk in miso until dissolved, then simmer 3 minutes. Stir in cream and mushroom mixture and cook until heated through, about 2 minutes. Serve with chicken and additional thyme, if desired.