Heat the dashi to a simmer in a saucepan. Add the mushrooms and simmer for 3 to 4 minutes. Soften the miso in a small bowl by mixing it with a few tablespoons of hot dashi. Pour the softened miso into the saucepan. Heat the soup, being careful not to boil, and add tofu. Pour into individual soup bowls and garnish with the green onions.
Combine the water and kelp in a large pot. Bring the water to a boil, removing the kelp just before water boils since it emits a strong odor. If the fleshiest part of the kelp is still tough, return it to the water for a few more minutes, along with a little cold water. (Soft kelp means sufficient flavor has been extracted.) Remove the kelp and bring stock to a full boil.
Add 1/4 cup cold water to bring the temperature down quickly and add the bonito flakes. Bring to a boil, and then immediately remove from the heat. (If bonito flakes boil too long, the stock becomes bitter.) Allow the flakes to settle to the bottom of the pan, then filter the stock through cheesecloth.
c.1997, M.S. Milliken & S. Feniger, all rights reserved