Heat the oil, add tomato paste and sugar, and cook for 30 seconds. Add white pepper and stir. Add the vegetable stock and bring to a boil.
When the stock comes to a boil, add the shiitake and wood ear mushrooms, bamboo shoots, carrots, tofu, soy sauce, and dark soy sauce. Bring to a rapid simmer. Stir in the doubanjiang and simmer for 2 minutes.
Combine the cornstarch and equal part water to make a cornstarch slurry. Stir in the cornstarch in the soup. Bubble for 2 minutes. If the soup becomes too thick, add water to thin it out. If the soup is not thick enough, add more cornstarch slurry to the soup.
Gently stir in the beaten egg. Once the egg is set, add the vinegar, and stir. Taste and adjust the vinegar and spice level to taste. Stir in the toasted sesame oil and green onions; serve immediately.