01 - Preheat oven to 350°F. Pat chicken pieces dry and season evenly with salt and freshly ground black pepper.
02 - Heat olive oil and 1 tablespoon butter in a Dutch oven over medium-high heat. Brown chicken in batches for 3 to 4 minutes per side. Transfer browned chicken to a plate.
03 - Add remaining butter to the pot. Sauté onions, carrots, and garlic for 4 to 5 minutes until vegetables are softened. Sprinkle flour over vegetables and cook 1 minute, stirring constantly.
04 - Pour in Calvados if using, allowing the liquid to reduce for 1 minute. Add dry cider and stir, scraping up any browned bits from the pot.
05 - Stir in chicken stock. Return browned chicken pieces to the pot, nestling them among the vegetables. Add apple slices, bay leaf, and thyme sprigs.
06 - Bring liquid to a gentle simmer. Cover and transfer to oven. Bake for 50 minutes.
07 - Uncover, stir in heavy cream, and continue baking uncovered for 10 to 15 minutes, until sauce thickens slightly and chicken is tender.
08 - Remove bay leaf and thyme sprigs. Taste for seasoning and adjust with salt and pepper if necessary.
09 - Serve casserole hot, garnished with fresh thyme if desired.