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.: You will notice an earthy aroma rising as the mushrooms rehydrate, and the soaking water darkens with umami. This soaking step unlocks intense mushroom flavor and softens the texture so they integrate well into the broth. A common mistake is rushing the soak, which leaves the mushrooms tough, so make time for the full soak. After squeezing them dry, reserve the soaking liquid, strain it to remove grit, and use it to deepen the broth, because that liquid concentrates the mushroom essence.
In a small bowl, add the cornstarch and mix in the water to make a smooth paste.: The slurry should be completely smooth and glossy, without lumps, which you can feel as a uniform texture when you stir. If the paste has lumps, the broth will get cloudy and grainy, so whisk until silky. Use cool water to disperse the starch evenly; warm water encourages premature thickening. A troubleshooting tip, if it appears too thick, add a splash more water and stir, because the right consistency will thicken the soup without making it gummy.
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.: As the onion sweats, you'll smell a gentle sweetness that builds the soup's foundation. When you pour in the chicken broth and reserved mushroom liquid, the steam carries savory, rounded notes. Bringing to a boil then reducing to a simmer extracts flavor and allows the mushrooms to finish tenderizing; you should see small bubbles and a calm simmer. Avoid a rolling boil for too long because it can toughen ingredients. If the soup tastes flat after simmering, a pinch of salt or a splash more soy sauce can bring it to life.
Stir in the cornstarch paste, and cook for 3 minutes to thicken. Add the beancurd, rice vinegar, soy sauce, kosher salt, and white pepper.: After you stir in the slurry you will notice the broth become glossy and slightly viscous within minutes, clinging to the tofu and mushrooms. This mouthfeel gives the soup a luxurious presence. Stir gently and keep the simmer steady, because high heat can break the starch down and make the texture thin again. A common error is adding the slurry too quickly, which can create lumps, so pour in slowly while whisking or stirring. Taste after adding the vinegar and soy sauce, and balance acid and salt gradually, because rice vinegar can brighten strongly.
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.: As the soup returns to a gentle boil you will see steam and small rolling bubbles, the perfect moment to add the beaten egg . Drizzling it through a whisk creates fine ribbons that cook instantly and add silky texture. The final stir of toasted sesame oil releases a fragrant nutty note that lifts the whole pot. Avoid pouring the egg too fast, which yields clumps instead of threads; a steady drizzle ensures elegance. Garnish with the chopped green onions right before serving so they stay bright and slightly crisp, adding a fresh contrast to the warm, rich broth.