Place mushrooms in a medium bowl and cover with boiling water.
Cover and let stand 10 minutes or until tender and drain. Thinly slice mushrooms and set aside.
Combine broth, 2 cups water, ginger, and garlic in a large saucepan over medium-high heat and bring to a boil.
Add mushrooms. Reduce heat, and simmer 5 minutes.
Add vinegar, soy sauce, pepper, and tofu and bring to a boil. Reduce heat, and simmer 5 minutes.
Combine remaining 1/4 cup water and cornstarch, stirring with a whisk.
Stir cornstarch mixture into broth mixture. Bring to a boil.
Reduce heat and simmer 3 minutes or until soup thickens slightly, stirring frequently.
Slowly pour egg whites into broth mixture in a steady stream, stirring constantly but gently with a wooden spoon. Remove from heat and stir in onions, cilantro, and sesame oil.