Combine chicken broth, 1 3/4 c. water, ginger and garlic in Dutch oven over medium-high heat; bring to a boil. Add mushrooms and reduce heat to a simmer; cook 5 minutes.
Add tofu, vinegar, red pepper flakes and soy sauce. Simmer, uncovered, 10 minutes.
Combine remaining 1/4 c. water wtih cornstarch; blend well. Stir cornstarch mixture into soup and simmer 3 minutes or until soup thickens. Slowly pour egg whites into broth, stirring constantly. Add green onions and sesame oil. Heat through and ladle into serving bowls.