Step 1

Heat 2 Tbsp. 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.

Step 2

Increase heat to medium-high and add oil and 1 Tbsp. 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.

Step 3

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.

Step 4

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.

Step 5

Meanwhile, heat remaining 2 Tbsp. 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.

Step 6

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.

Step 7

DO AHEAD: Chicken and apples can be cooked 1 day ahead. Cover and chill. Cook mushrooms and finish sauce just before serving.