Asian Chicken Noodle Soup

Name: Asian Chicken Noodle Soup
Ingredients: 2 boneless (skinless chicken breasts (about 1 pound), thinly sliced); 2 medium carrots (julienned); 4 green onions (thinly sliced (white and green parts separated)); 1/4 cup mirin; 1 tablespoon fresh ginger (minced); 1/2 small cabbage (thinly sliced (about 2 cups)); 1 cup mushrooms (sliced (shiitake or button)); 6 cups chicken broth; 8 ounces noodles (udon, soba, or rice noodles); 2 tablespoons soy sauce; 1 tablespoon sesame oil; Salt and pepper to taste