Heat the oil in your wok over medium heat. Add the ginger, and cook for 1 minute, or until fragrant and beginning to crisp at the edges. Increase the heat to high and add the chicken. Cook the chicken for 3-5 minutes, or until lightly golden.
Add the soaked shiitakes and wood ear mushrooms. Stir-fry for 2 minutes.
Add the 1 1/4 cup of the water you used to soak the mushrooms. Also add the Shaoxing wine, sugar, light soy sauce, and dark soy sauce. Mix everything well, and bring the liquid to a boil.
Reduce the heat to low, cover the wok, and simmer for 40-45 minutes, or until most of the liquid has cooked off. Make sure to stir occasionally, and remember to keep the heat low (add additional water if it dries out too quickly).
Finally, stir in the scallions until they're wilted. Serve!