1. Place the water, chicken, dried mushrooms, ginger and scallions in a large pot and bring to a boil. Reduce the heat to low, then 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 allow them to 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 season with additional salt if needed.
4. Remove the chicken parts from the soup. Strain the broth through a fine mesh strainer, discarding the scallions and ginger. Skim to remove excess fat from the broth.
5. To serve, add the baby bok choy to the simmering, strained broth 15 minutes before serving and cook until tender. Remove the bok choy from the broth and quarter them lengthwise. Serve one or two pieces, along with a few of the reserved mushroom slices, atop each soup bowl.