1. Combine first 3 ingredients in a 2-quart sauce-pan; bring to a boil. Add chicken, and simmer, uncovered, 10 minutes. Add bamboo shoots; simmer 5 minutes. Add vinegar and next 3 ingredients; bring to a boil. Drizzle egg white into soup, stirring constantly (lacy strands will form). Stir in green onions and snow peas.
2. Combine cornstarch and water; stir into soup mixture. Bring to a boil; boil 1 minute, stirring gently. Serve immediately.