Preheat oven to 350˚F.

In a large, deep, ovenproof skillet over medium heat, heat 2 tablespoons of the olive oil. Add the mushrooms and sprinkle with salt and pepper. Cover and cook, stirring occasionally, until tender and golden brown, about 5 minutes. Transfer to a large bowl.

Heat the last two tablespoons of oil in the skillet over medium-high heat. Sprinkle the chicken with salt and pepper and place skin-side-down in the skillet, and cook about 4 minutes, until the skin-side is browned, Flip and brown the other side, about 3 minutes. Transfer the chicken to a plate, and pour off all but two tablespoons of fat from the skillet. Cook the chopped apples in the skillet over medium-high heat for about 2 minutes, until browned. Stir in the shallots and garlic and reduce heat to low. Cook 3 minutes, stirring occasionally, until softened. Stir in the flour, then the Calvados. Cook for 1 minute. Stir in the chicken stock, apple cider vinegar, caraway seeds, and crushed red pepper flakes. Bring to a simmer.

Return the chicken pieces, skin-side-up, to the skillet with any juices. Braise the chicken in the oven for 20 minutes, until the breast meat is just cooked, and transfer the breast pieces to a large (ovenproof) platter. Continue to cook the rest of the chicken for 10 minutes. Add to the platter and keep warm.

Boil the pan juices with the apples over medium-high heat until it begins to thicken, 5-6 minutes. Fold in the cooked mushrooms. Remove skillet from heat, and stir in the butter, a tablespoon at a time. Season the sauce to taste with salt and pepper and stir in half of the chives. Serve the chicken with the sauce spooned over it, and garnish with the remaining chives.