Begin by placing the dried shiitake mushrooms in a bowl of hot water and let them rehydrate for 15 to 20 minutes, stirring occasionally. Once softened, drain thoroughly and slice them into thin strips.
In a small bowl, whisk together the soy sauce, rice vinegar, white pepper, and sesame oil. This will be your flavor base—set it aside for later use.
In a separate bowl, combine the cornstarch and water to create a slurry. Stir until smooth and lump-free, and keep it aside.
Heat vegetable oil in a large saucepan over medium-high heat. Add the minced ginger, green onions, and rehydrated mushroom slices. Sauté the mixture for about 2 minutes until aromatic and slightly softened.
Pour in the chicken broth and bring the mixture to a gentle simmer. Lower the heat to medium and let it cook for 5 minutes, allowing the broth to absorb the umami flavors.
Add the grated carrot, red bell pepper, bamboo shoots, tofu cubes, and the prepared soy-vinegar mixture. Let the soup simmer for another 5 minutes to allow the vegetables to soften and the flavors to meld.
Increase the heat to medium-high and bring the soup to a rolling boil. Slowly pour in the beaten eggs in a thin, steady stream while stirring the soup in one direction to create delicate egg ribbons. Continue stirring until the soup returns to a boil.
Re-stir the cornstarch slurry to ensure it’s fully blended, then slowly drizzle it into the soup while stirring constantly. Cook for another 1 to 2 minutes until the broth thickens to a silky consistency.
Turn off the heat, adjust seasoning to your taste if needed, and serve hot.