Heat the light sesame oil (or peanut oil if using as a substitute) in a skillet. Add the ginger, garlic, and green onions and stir-fry for 30 seconds. Then add the mushrooms, along with their soaking water. Cook for 2-3 minutes, then add the broccoli and cooked noodles. Pour the soy sauce/ mirin mixture over the vegetables and noodles and make sure everything is heated.