Place the dried shiitake and wood ear mushrooms (if using) in a small bowl. Cover with hot water and soak for about 30 minutes. Squeeze the mushrooms dry, reserving the soaking water. Cut off the shiitake mushroom stems, discard, and slice the mushroom caps. Trim away any tough stems from the wood ears, and finely chop.
In a small bowl, add the cornstarch and mix in the water to make a smooth paste.
In a 3 quart saucepan, heat the oil over medium. Add the onion and cook until soft, 3-5 minutes. Add the chicken broth, mushrooms, wood ears, and soaking water. Bring to a boil, then reduce to a simmer for 15 minutes.
Stir in the cornstarch paste, and cook for 3 minutes to thicken. Add the beancurd, light soy sauce, kosher salt, and white pepper.
Bring the soup to a lazy boil then drizzle in the beaten egg letting it drop through a whisk to make threads in the soup. Stir in the sesame oil and serve with chopped green onion on top.