Preheat the oven to 350°. In a large, deep ovenproof skillet, heat 2 tablespoons of the olive oil. Add the mushrooms and season with salt and pepper. Cover and cook over moderate heat, stirring a few times, until browned and tender, about 5 minutes. Transfer the mushrooms to a large bowl.
Heat the remaining 2 tablespoons of olive oil in the skillet. Season the chicken pieces with salt and pepper and add to the skillet, skin side down. Cook over moderately high heat until browned, about 4 minutes. Turn the chicken and brown the other side, about 3 minutes. Add the chicken to the mushrooms and pour off all but 2 tablespoons of fat from the skillet.
Add the chopped apples to the skillet and cook over moderately high heat until browned, about 2 minutes. Add the shallots and garlic and cook over low heat, stirring a few times, until softened, about 3 minutes. Stir in the flour. Add the Calvados and cook for 1 minute. Stir in the chicken stock, cider, caraway seeds and crushed red pepper and bring to a simmer.
Return the chicken pieces to the skillet, skin side up, along with any accumulated juices. Braise the chicken in the oven for about 20 minutes, until the breast meat is just cooked. Transfer the breast pieces to a large ovenproof platter. Continue braising the remaining chicken until cooked through, about 10 minutes longer. Add to the platter and keep warm.
Boil the pan juices and apples over moderately high heat until slightly thickened, about 6 minutes. Add the mushrooms. Remove the skillet from the heat and swirl in the butter, 1 piece at a time. Season the sauce with salt and pepper and add half of the chives. Spoon the sauce over and around the chicken. Garnish with the remaining chives and serve.