01 - Preheat oven to 350°F.
02 - Pat chicken pieces dry. Season 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 on both sides in batches, about 3-4 minutes per side; set aside.
04 - In the same pot, add remaining butter. Sauté onions, carrots, and garlic for 4-5 minutes until softened. Sprinkle with flour and cook 1 minute more.
05 - Deglaze with Calvados (if using) and then add cider. Stir, scraping any browned bits from the bottom.
06 - Add the chicken stock, then return browned chicken pieces to the pot. Add apple slices, bay leaf, and thyme sprigs.
07 - Bring to a simmer. Cover, transfer to oven, and bake for 50 minutes.
08 - Uncover, stir in heavy cream, and bake uncovered for another 10-15 minutes until the sauce thickens slightly and chicken is tender.
09 - Remove bay leaf and thyme sprigs. Taste and adjust seasoning.
10 - Serve hot, garnished with additional thyme if desired.