Place the walnuts, sage, thyme, and garlic into a food processor. Pulse until the mixture is coarse and crumbly, stopping before it becomes a paste.

Transfer the walnut mixture to a wide, shallow dish. Add the flour and mix well so everything is evenly combined.

Pat the chicken dry with paper towels. Rub with 1 tablespoon of olive oil and season generously with salt and pepper. Press each piece into the walnut mixture, making sure the coating sticks well. Cover and refrigerate for at least 1 hour to help the crust firm up.

Preheat the oven to 350°F. Heat the remaining olive oil and butter in a large skillet over medium-high heat. When hot, sear the chicken on all sides until golden. Some crust may fall into the pan—this adds flavor to the sauce later.

Transfer the chicken to a lightly greased baking dish. Bake for 15–20 minutes, or until the internal temperature reaches 165°F. Remove from the oven, loosely cover with foil, and let rest.

Return the skillet to medium heat. Add the chopped onion to the pan drippings and cook until softened. Add a small amount of butter if the pan is dry.

Sprinkle in the flour and stir for about 1 minute until lightly golden. Slowly whisk in the apple cider and chicken broth until smooth. Season with salt and pepper, then add the bay leaf and apple cider vinegar.

Let the sauce simmer gently until thickened, stirring occasionally. Remove the bay leaf and strain if a smoother sauce is desired.

Slice the chicken and spoon the warm apple cider pan sauce over the top. Serve immediately.