01 - Preheat oven to 350°F.
02 - Pat chicken pieces dry with paper towels. Season generously on all sides with salt and pepper.
03 - In a large Dutch oven or ovenproof casserole, heat olive oil and 1 tbsp butter over medium-high heat. Brown chicken pieces on both sides in batches, about 3-4 minutes per side. Transfer browned chicken to a plate and set aside.
04 - Add remaining butter to the same pot. Sauté onions, carrots, and garlic for 4-5 minutes until softened. Sprinkle flour over vegetables and cook for 1 minute, stirring constantly.
05 - Deglaze with Calvados if using, then add cider. Stir well, scraping up any browned bits from the bottom of the pot.
06 - Add chicken stock and return browned chicken pieces to the pot. Add apple slices, bay leaf, and thyme sprigs. Bring to a gentle simmer.
07 - Cover the pot and transfer to the preheated oven. Bake for 50 minutes.
08 - Uncover the pot and stir in heavy cream. Return to oven and bake uncovered for another 10-15 minutes until sauce thickens slightly and chicken is tender.
09 - Remove and discard bay leaf and thyme sprigs. Taste the sauce and adjust seasoning with salt and pepper if needed.
10 - Serve hot, garnished with additional fresh thyme if desired.