Place mushrooms in a medium bowl; cover with boiling water. Cover and let stand 10 minutes or until tender; drain. Thinly slice mushrooms; set aside.
Combine broth, 2 cups water, ginger, and garlic in a large saucepan over medium-high heat; bring to a boil. Add mushrooms. Reduce heat, and simmer 5 minutes. Add vinegar, soy sauce, pepper, and tofu; 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; 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; stir in onions, cilantro, and sesame oil. Drizzle with chili oil, if desired.