01 - Preheat oven to 350°F.
02 - Pat chicken pieces dry with paper towels. Season generously on all sides with salt and black 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 and fragrant. Sprinkle flour over vegetables and cook for 1 minute, stirring constantly.
05 - Add Calvados (if using) and stir to scrape up any browned bits from the bottom. Pour in the cider and continue stirring until well combined.
06 - Add chicken stock to the pot. Return browned chicken pieces along with any accumulated juices. Add apple slices, bay leaf, and thyme sprigs. Stir gently to combine.
07 - Bring the mixture to a simmer. Cover the pot and transfer to the preheated oven. Bake for 50 minutes.
08 - Remove the pot from the oven. Uncover and stir in heavy cream. Return to the oven and bake uncovered for an additional 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 fresh thyme sprigs if desired.