You can use any type of miso paste for this soup. Red miso offers the strongest flavor. For milder results, try white miso.
1. Combine shiitake mushrooms, kombu, ginger, soy sauce, sake, sugar (if using), and 5 cups water in saucepan; bring to a boil. Cover, reduce heat to medium-low, and simmer 10 minutes. Strain out solids, and discard shiitake mushrooms and ginger. Slice kombu, and set aside. Return liquid to saucepan.
2. Add button mushrooms and leek to broth, and bring to a boil. Cover, reduce heat to medium-low, and simmer 5 minutes. Add sweet potato, green beans, and sliced kombu; simmer 5 to 7 minutes more, or until sweet potato is tender.
3. Whisk together miso and 1 cup warm broth from soup in bowl until miso has dissolved. Stir miso mixture into soup, and heat over medium-low heat until miso “blooms.” Stir in tofu, and heat 1 to 2 minutes (do not boil). Serve garnished with green onions and drizzled with sesame oil (if using).