Pat the chicken dry and season well with salt and pepper. Heat the olive oil in a Dutch oven or braiser over medium-high heat.
Add the chicken, skin-side down first, and sear until golden on both sides, 4–5 minutes per side. Transfer to a plate.
Lower the heat to medium and add half of the butter cube (2 tablespoons) along with the shallots and garlic. Cook 3–4 minutes, stirring often, until softened and fragrant.
Stir in the Dijon, then pour in the hard cider and chicken stock, scraping up the brown bits.
Nestle the chicken back into the pot, skin-side up. Tuck in the thyme, sage, and apple wedges.
Cover with a lid and transfer to a 350°F oven for 40 minutes.
Uncover and cook another 20 minutes, until the chicken is tender and the skin is golden. For an extra-crispy finish, broil 1–2 minutes.
Transfer the chicken and apples to a serving platter.
Simmer the liquid on the stove uncovered for 5–7 minutes until slightly reduced, then stir in the remaining 2 tablespoons butter until glossy. Spoon the sauce and apples over the chicken and serve with mashed potatoes.
