How to Make It
In a medium saucepan, toast the walnuts over moderate heat until lightly golden, about 5 minutes. Chop the nuts and transfer to a small bowl. In the same saucepan, heat the olive oil. Add the onion and cook over moderate heat, stirring occasionally, until golden, about 3 minutes. Add the lentils, bay leaf, balsamic vinegar and 1 1/2 cups of water and bring to a simmer. Cover and cook over low heat for 15 minutes. Uncover and cook, stirring occasionally, until the liquid is absorbed and the lentils are just tender, 15 minutes longer. Stir in the butter and mustard and season with salt and pepper. Keep warm.
Preheat the oven to 400°. In a large cast-iron skillet, heat the olive oil. Season the chicken with salt and pepper and set breast side down in the skillet. Cover with foil and another large cast-iron skillet. Cook the chicken over moderate heat until golden, 8 to 10 minutes. Flip the chicken halves and roast in the oven, uncovered, until golden and an instant-read thermometer inserted in the inner thigh registers 165°, about 30 minutes. Transfer the chicken to a cutting board and let rest for 5 minutes. Whisk the butter into the pan juices and season with salt and pepper. Strain the jus and keep warm.
In a small bowl, whisk the walnut oil with the sherry vinegar and parsley. Season with salt and pepper.
Carve the chicken. Stir the toasted walnuts into the lentils and spoon onto plates. Top with the chicken. Drizzle with the vinaigrette and serve the chicken jus on the side.