Toasted walnuts and chopped-radicchio salad top chicken breasts flavored with walnut oil and lemon juice.
Heat oven to 350 degrees. Place walnuts on a baking sheet; toast until golden brown and fragrant, 8 to 10 minutes. In a bowl, combine lemon juice and 1/2 teaspoon walnut oil. Brush over chicken breasts; set aside.
Fill a bowl with water and ice; set aside. Bring a pot of water to a boil; cook beans until tender, about 1 1/2 minutes. Drain; transfer to ice bath to cool. Drain again; cut beans into 1/2-inch pieces. Combine beans, radicchio, and fennel in a large bowl. Add vinegar, remaining 2 tablespoons walnut oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Toss; set aside.
Heat a grill or grill pan coated with olive-oil cooking spray until very hot. Sprinkle chicken with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Grill until browned, about 3 minutes. Turn; continue cooking until browned and cooked through, 2 to 3 minutes more. Serve topped with reserved radicchio salad; garnish with walnuts.
Place the chicken-breast halves between two layers of plastic wrap before pounding them.