01 - Set oven to 350°F (180°C) to preheat.
02 - Pat chicken pieces dry and generously season with salt and freshly ground black pepper on all sides.
03 - Heat olive oil and 1 tablespoon butter in a large Dutch oven or ovenproof casserole over medium-high heat. Brown the chicken in batches, 3 to 4 minutes per side. Transfer browned pieces to a plate.
04 - Add the remaining butter to the same pot. Sauté onions, carrots, and minced garlic for 4 to 5 minutes until just softened. Sprinkle flour over the vegetables and cook for 1 minute, stirring continuously.
05 - Carefully pour in Calvados if using, allowing it to sizzle for a moment. Add the dry cider and stir, scraping up any browned bits from the bottom of the pot.
06 - Pour in chicken stock. Return browned chicken pieces to the pot. Arrange apple slices, bay leaf, and thyme sprigs between the chicken pieces.
07 - Bring the mixture to a gentle simmer. Cover the pot, transfer to the preheated oven, and bake for 50 minutes.
08 - Uncover the pot. Stir in the heavy cream and continue baking uncovered for 10 to 15 minutes until the sauce thickens slightly and the chicken becomes fully tender.
09 - Remove bay leaf and thyme sprigs. Taste the sauce and adjust salt and pepper as desired.
10 - Serve hot, garnished with fresh thyme if available.