Bring about 2 cups of water to a boil. Place the dried mushrooms in a small bowl. Cover with the boiling water and place something like a small saucer on top to keep them submerged. Allow the mushrooms to soak for 30 minutes.
Fill a small saucepan with water and bring to a boil. Gently lower the eggs into the pot. As soon as the water starts to bubble again, set a timer for six minutes. After cooking, drain and immediately flush eggs well with cold water or place eggs into an ice bath. When eggs are cool, peel and set aside.
Slice scallions into 2" lengths and separate the white and green parts. Crush the white part slightly with the side of knife. Peel the ginger and slice into 1/4" slices. Set aside.
Slice the chicken into 1/2" thick slices. Drain the mushrooms, reserving slightly more than 1 cup (200 ml) of the mushroom soaking liquid. Remove the stems and slice mushrooms into thick 1/4" slices. Set aside.
Heat a wok over high heat. When hot, add the oil, swirl, then add the chicken. Stir-fry for a few minutes until chicken is almost cooked through and has a little color. Add the scallion whites and ginger. Stir-fry for 30 seconds or until fragrant.
Add the shaoxing wine, stir a few times, then add the mushrooms, mushroom soaking water, sugar, dark soy and salt. Nestle in the eggs, bring to a boil and cover the pan. Reduce heat to a simmer and cook for 30 minutes. Gently stir the meat and rotate the eggs occasionally. This will ensure the eggs are evenly colored.
After cooking, reduce the liquid, if desired, by simmering over high heat without the lid on for two minutes. Add the scallion greens and sesame oil. Serve with plain, steamed rice.