- 5 tablespoons unsalted butter, divided
- 3 firm, tart apples (such as Pink Lady or Braeburn; about 1 pound), peeled, cored, quartered
- 1 tablespoon olive oil
- 2 3 1/2–4-pound chickens, quartered
- Kosher salt, freshly ground pepper
- 1 leek, white and pale-green parts only, halved lengthwise, sliced crosswise 1/4″ thick
- 2 shallots, finely chopped
- 1/4 cup Calvados (apple brandy)
- 3/4 cup apple cider
- 4 sprigs thyme
- 2 bay leaves
- 1/2 cup low-sodium chicken broth
- 1/2 pound crimini (baby bella) mushrooms, trimmed, halved
- 1/2 cup crème fraîche
- 1 large egg yolk
- Heat 2 tablespoons butter in a large heavy pot over medium heat. Add apples and cook, turning occasionally, until golden in spots, 10–12 minutes. Transfer apples to a plate and set aside.
- Increase heat to medium-high and add oil and 1 tablespoon butter to pot. Season chicken with salt and pepper and, working in batches, cook until browned, about 5 minutes per side. Transfer chicken to another plate; set aside.
- Add leek and shallots to pot; cook, stirring often, until softened, about 4 minutes. Remove pot from heat, add Calvados, and ignite with a long match or lighter. After flames die down, return pot to heat and add cider. Bring to a boil, reduce heat, and simmer until slightly reduced, about 3 minutes.
- Return reserved chicken to pot and add thyme, bay leaves, and broth. Bring to a boil; reduce heat, cover pot, and simmer, adding reserved apples back to pot halfway through, until chicken is cooked through, 20–25 minutes.
- Meanwhile, heat remaining 2 tablespoons butter in a large skillet over medium-high heat. Add mushrooms and cook, tossing occasionally, until browned and softened, 6–8 minutes; season with salt and pepper. Transfer mushrooms to a plate.
- Whisk crème fraîche and egg yolk in a small bowl. Using a slotted spoon, transfer chicken and apples to a baking sheet and remove pot from heat. Whisk crème fraîche mixture into cooking liquid in pot. Gently mix in chicken, apples, and mushrooms.
- DO AHEAD: Chicken and apples can be cooked 1 day ahead. Cover and chill. Cook mushrooms and finish sauce just before serving.