01 - Preheat oven to 350°F (180°C).
02 - Pat chicken pieces dry with paper towels. Season generously on all sides with salt and freshly ground black pepper.
03 - In a large Dutch oven or ovenproof casserole, heat olive oil and 1 tablespoon butter over medium-high heat. Brown chicken pieces on both sides in batches, cooking 3-4 minutes per side until golden. 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, stirring constantly, for 1 minute to cook out the raw flour taste.
05 - Add Calvados if using, followed by the cider. Stir well, scraping up any browned bits from the bottom of the pot with a wooden spoon.
06 - Pour in chicken stock and return browned chicken pieces to the pot. Add sliced apples, bay leaf, and thyme sprigs. Stir to combine all ingredients.
07 - Bring the mixture to a simmer. Cover the pot and transfer to the preheated oven. Bake for 50 minutes until chicken is nearly tender.
08 - Remove the pot from oven and uncover. Stir in heavy cream until well combined. Return to oven and bake uncovered for an additional 10-15 minutes until sauce has thickened slightly and chicken is completely tender.
09 - Discard bay leaf and thyme sprigs. Taste the sauce and adjust seasoning with additional salt and pepper if needed.
10 - Serve hot, garnished with additional fresh thyme sprigs if desired. Pairs well with mashed potatoes, crusty baguette, or buttered egg noodles.