Heat oven to 375°F.
Heat 2 TBSP oil in a large oven-safe skillet on medium-high.
Add mushrooms and a pinch of salt and cook, tossing occasionally, 5 minutes. Add shallot and cook, stirring occasionally, until mushrooms are golden brown, 2 to 3 minutes; transfer to a plate.
Add apples and butter to the skillet and cook, tossing occasionally, until beginning to turn golden brown, 2 to 3 minutes; transfer to a second plate and wipe skillet clean.
Return skillet to medium. Rub chicken with remaining TBSP olive oil, season with 1/2 tsp each salt and pepper, and cook, bone side up first, until browned on all sides, 10 to 12 minutes, drain excess fat.
Turn chicken skin side up, add wine, and cook 1 minute, then add broth and thyme.
Return apples to the skillet, transfer skillet oven, and bake until chicken is cooked through (165°F), 10 mins.
Transfer chicken and apples to a plate, discard thyme, and return skillet to medium heat. Whisk in cream and mustard.
Then fold in mushroom mixture and cook until heated through, about 2 minutes.
Serve with chicken and apples and additional thyme if desired.
