Place the dried wood ear mushrooms in a heatproof bowl and cover with boiling water. Let stand for 10 minutes, then drain, rinse well, trim away any tough stem pieces, and slice thinly.
In a small bowl, toss the pork with 1 teaspoon soy sauce and 1 teaspoon cornstarch until coated.
Set a large soup pot over medium-high heat. Add the chicken broth, soaked wood ear mushrooms, shiitake mushrooms, bamboo shoots, ginger, remaining 2 tablespoons soy sauce, black vinegar, and white pepper. Bring to a boil.
Add the coated pork. Reduce the heat to maintain a gentle simmer and cook for 3 to 4 minutes, until the pork is cooked through. Add the tofu and simmer gently for 1 minute.
In a small bowl, whisk the remaining 3 tablespoons cornstarch with the cold water until completely smooth. Stir the slurry into the simmering soup and cook for 1 to 2 minutes, stirring gently, until the broth is lightly thickened.
Stir the soup in a slow circle with a spoon. Gradually pour in the beaten egg in a thin stream, then stop stirring and let the egg ribbons set for about 30 seconds.
Remove the pot from the heat. Stir in the sesame oil and scallions. Taste and add a little more black vinegar or white pepper if desired. Serve hot, with chili oil at the table for anyone who wants extra heat.