Preheat oven to 400 degree F. Place each chicken breast half between two pieces of plastic wrap. Pound breasts lightly into a rectangle about 1/8 inch thick. Remove plastic wrap. Spread one-fourth of the cheese in center of each breast; sprinkle with green onion.
Fold in sides of chicken; roll up from one of the short ends, pressing edges to seal. Secure with wooden toothpicks, if necessary.
In a shallow dish combine egg, water, salt, and pepper. Beat with a whisk or fork until well blended; set aside. In another shallow dish combine bread crumbs and walnuts. Coat chicken rolls with flour. Dip chicken in egg mixture, then in walnut mixture to coat all sides. Place chicken, seam-side down, in a 2-quart square baking dish. Drizzle with melted margarine or butter.
Bake, uncovered, for 20 to 25 minutes or until chicken is tender and no longer pink inside. Top the chicken rolls with mixed baby greens. Return to oven. Bake about 1 minute more or until the greens just begin to wilt. Remove from oven. Sprinkle greens with the balsamic vinegar. Makes 4 servings.