01 - Preheat oven to 350°F.
02 - Pat chicken pieces dry and season generously on all sides with salt and pepper.
03 - In a large Dutch oven, heat olive oil and 1 tbsp butter over medium-high heat. Brown chicken pieces on both sides in batches, 3-4 minutes per side. Transfer to plate and set aside.
04 - Add remaining butter to the pot. Sauté onions, carrots, and garlic for 4-5 minutes until softened. Sprinkle flour over vegetables and cook for 1 minute, stirring constantly.
05 - Add Calvados if using, then pour in cider. Stir well, scraping browned bits from bottom of pot.
06 - Add chicken stock and return browned chicken to pot. Add apple slices, bay leaf, and thyme sprigs. Bring mixture to a simmer.
07 - Cover pot and transfer to oven. Bake for 50 minutes.
08 - Uncover pot, stir in heavy cream, and continue baking uncovered for 10-15 minutes until sauce thickens slightly and chicken is tender.
09 - Remove and discard bay leaf and thyme sprigs. Taste sauce and adjust seasoning with salt and pepper as needed.
10 - Serve hot, garnished with fresh thyme sprigs if desired.