When the mushrooms are soft, remove the mushrooms and lily buds and reserve the soaking liquid. Remove any tough pieces of the mushrooms and slice into strips. Discard the tough ends of the lily buds and cut in half. Combine and set aside.
Strain the reserved mushroom soaking liquid through a fine-mesh sieve lined with a dampened paper towel. Bring to a boil in a soup pot along with the chicken stock.
Add the mushrooms and tiger lilies, reduce heat, and simmer for 3 minutes.
While the soup is simmering, in a cup, thoroughly combine 3 Tbsp cornstarch and 3 Tbsp cold water. Whisking constantly, pour it into the simmering soup and allow it to simmer until slightly thickened.
Add the meat and tofu as well as 3/4-1 tsp ground black pepper.
Bring back to a simmer then stir into the soup in a wide circle: 1 egg, well-beaten.
Remove from heat and add: 1 Tbsp sesame oil.
Serve and garnish with sliced green onion and more rice vinegar and chili oil for more heat.