In a medium bowl, gently toss the tofu cubes with 1 tablespoon (15 ml / 0.5 fl oz) of soy sauce and the toasted sesame oil.
Set aside to marinate.
In a separate small bowl, pour the boiling water over the dried wood ear mushrooms.
Let them stand for 20 minutes to rehydrate, then drain well.
In a large saucepan or Dutch oven, combine the vegetable broth, drained mushrooms, bamboo shoots, shredded carrot, rice vinegar, the remaining 2 tablespoons (30 ml / 1 fl oz) of soy sauce, sugar, fresh ginger, chili oil, and white pepper.
Bring the mixture to a rolling boil over medium-high heat.
Reduce the heat to a simmer, cover, and cook for 2 minutes to let the flavors meld. Gently stir in the marinated tofu mixture and simmer for 1 more minute.
In a small bowl, whisk together the cornstarch and cold water to create a slurry. Slowly pour the slurry into the simmering soup, stirring constantly until it thickens and becomes bubbly.
Continue to cook and stir for 2 more minutes.
Remove the saucepan from the heat.
While gently stirring the soup in one direction, slowly pour the beaten egg in a thin, steady stream. This will create beautiful, delicate egg ribbons.
Stir in the sliced green onions and serve immediately.