Place the dried wood ear and shiitake mushrooms in separate bowls and cover with hot water. Let them soak for 20 to 30 minutes, until fully softened. Squeeze out the excess water, trim away any tough stems, and slice thinly.
Bring the chicken broth to a boil in a large pot or Dutch oven over medium-high heat. Stir in the ginger, soy sauce, salt, and sugar.
Add the sliced mushrooms, bamboo shoots, and tofu strips to the pot. Reduce the heat to medium and let it simmer for 8 to 10 minutes, until the bamboo shoots have softened slightly and the flavors have started to meld.
Give the cornstarch slurry a quick stir (cornstarch settles fast), then drizzle it into the simmering soup while stirring constantly. Let the soup come back to a gentle boil for 1 to 2 minutes until the broth turns lightly glossy and coats the back of a spoon.
Stir in the black vinegar. Taste and adjust until the soup is noticeably tangy. If your vinegar is mild or you like a bolder sour note, add another teaspoon or two.
With the soup at a gentle simmer (not a rolling boil), slowly drizzle the beaten eggs into the pot while stirring the soup in slow circles with a chopstick or spoon. The eggs will set into soft, feathery ribbons almost immediately.
Turn off the heat. Stir in the white pepper, sesame oil, and the scallion whites. Ladle into warm bowls, top with the scallion greens, and serve right away.