1. Place the water, chicken, dried mushrooms, ginger and green onions in a large pot and bring to a boil. Reduce the heat to low and simmer for 1½ hours, skimming the surface periodically to remove any impurities (foam). Add fresh water to the pot as needed to keep the liquid level constant.
2. After one hour of simmering, extract the mushrooms from the pot and let them cool a bit. Remove and discard their stems, then slice the mushroom caps thinly. Set them aside—they will be used later as garnish.
3. Add the rice wine or sake and 2 teaspoons salt, then simmer the broth for another 30 minutes. Stir, taste, and add more salt if needed.
4. Remove the chicken parts from the soup. Strain the broth through a fine mesh strainer, discarding the onions and ginger. Skim the broth to remove excess fat, then return the broth to a simmer.
5. 15 minutes before serving, add the baby bok choy (if using) to the simmering, strained broth and cook until crisp-tender. Lift the bok choy from the broth and quarter them lengthwise. Top each soup bowl with 1 or 2 pieces of bok choy and a few reserved mushroom slices.