1. Soak adzuki beans in 4 cups cold water 5 hours, or overnight. Drain, and reserve soaking liquid.
2. Coat large saucepan or Dutch oven with cooking spray, and heat over medium-low heat. Add white parts of green onions, sprinkle with salt, and cover. Cook 5 to 7 minutes, or until onions are soft. Add shiitake mushrooms and adzuki beans. Add enough water to soaking liquid to make 4 cups, and stir into bean mixture. Cover, and bring to a boil.
3. Reduce heat to medium-low, and simmer 45 minutes, or until beans are beginning to be tender, adding a little water if necessary. Fold in potatoes, cover, and cook 15 minutes more, or until potatoes are tender and beans are soft, but not mushy.
4. Ladle 1 cup broth from bean mixture into measuring cup. Stir in miso paste with fork until dissolved. Stir miso mixture into bean mixture, and simmer 2 to 3 minutes more, or until heated through. Sprinkle with chopped green parts of green onions before serving.