In separate bowls, soak the lily buds and mushrooms in very warm water until soft, about 30 minutes.
Meanwhile, rinse and julienne the bamboo shoots, then the tofu. Slice the pork or chicken into a similar pile of matchsticks, and work ½ tablespoon of cornstarch into it.
Julienne the fungus when it is soft enough, and follow suit with the lily buds after trimming off any hard stems.
In a small bowl, stir 3 tablespoons cold water into 1 tablespoon of cornstarch. Set aside.
In a medium bowl, whisk together the eggs and 1 tablespoon cool water. Set aside.
Bring 4 cups of water to a boil in a soup pot. Add julienned bamboo shoots and black fungus. Blanch for 2 minutes. Remove with a strainer and set aside. Add the julienned pork or chicken to the same blanching water and stir and cook gently for 2 minutes. Remove with a strainer and set aside. Discard the blanching water.
Bring the chicken stock to a simmer and add the bamboo shoots, black fungus, lily buds, pork or chicken, and tofu. When it returns to a simmer add soy sauce and white pepper, seasoning with a light hand to start.
Remix the cornstarch slurry and stir in about half to the simmering soup. Add more only if the soup remains too thin. (How do you know? Think “silken.” One of the beauties of making this at home is that you can avoid the gloppy nature of many restaurant soups.)
Drizzle the beaten eggs into the soup while stirring slowly. (Stirring too quickly or vigorously will cloud the soup, rather than create the lacy strings of egg desired.) Turn off the heat.
Add vinegar and salt to taste. Use your judgment to balance out the hot (white pepper), sour (vinegar), and umami (soy sauce) with the salt. Take an extra moment here to really taste.
Add the sesame oil, ladle into small bowls and garnish with scallion greens. Furnish the table with more white pepper for those who like it.