Season the chicken with salt; place the broth (or water) and chicken in a small stockpot. Bring to a simmer over medium heat, cover, and cook until chicken is just done, about 10 minutes. Remove breasts; cool, shred, and set aside. Reserve broth for future use.
Place basil, vinegar, garlic, oil, and salt to taste in blender. Blend until smooth and set aside.
In a large bowl, toss arugula, radicchio, lettuce, fennel, red pepper, onion, walnuts, 1/2 cup salad dressing, and salt and pepper to taste. Place three endive leaves in a three-pointed star with tips facing out on four individual dinner plates or wide shallow bowls. Place equal mounds of salad in center. Arrange chicken on top.