Bring stock, mushrooms and bamboo shoots to boil in large saucepan. Reduce heat to medium-low; simmer 5 minutes. Stir in vinegar, soy sauce, sugar, red pepper and white pepper.
Mix cornstarch and water in small bowl until smooth. Stir into saucepan with wire whisk. Gradually whisk in egg white; simmer until cooked through.
Ladle into soup bowls. Sprinkle with green onion and sesame seed.