01 - Set oven temperature to 350°F (180°C).
02 - Pat chicken pieces dry with paper towels. Season generously on all sides with salt and freshly ground black pepper.
03 - Heat olive oil and 1 tablespoon butter in a large Dutch oven or ovenproof casserole over medium-high heat. Sear chicken in batches until golden on both sides, about 3–4 minutes per side. Transfer browned chicken to a plate and set aside.
04 - Add remaining butter to the pot. Sauté onions, carrots, and garlic for 4–5 minutes until vegetables are softened. Sprinkle with flour and cook, stirring, for 1 minute to form a light roux.
05 - Pour in Calvados, if using, to deglaze the pan, scraping up browned bits. Add dry cider and mix well.
06 - Stir in chicken stock. Return chicken pieces to the pot along with apple slices, bay leaf, and thyme sprigs.
07 - Bring mixture to a simmer. Cover pot, transfer to preheated oven, and bake for 50 minutes.
08 - Uncover, pour in heavy cream, and continue baking uncovered for 10–15 minutes until sauce thickens slightly and chicken is fully tender.
09 - Remove bay leaf and thyme. Taste sauce and adjust seasoning with salt and pepper if needed. Serve hot, garnished with extra thyme if desired.