Combine chicken broth, water, bamboo shoots, shiitake mushrooms, garlic, ginger root, soy sauce, and red pepper flakes in a saucepan over medium-high heat. Bring to a boil. Reduce heat to low; cover and simmer for 5 minutes. Remove lid and add tofu. Return to a boil; slowly drizzle in egg while stirring. Stir in cornstarch mixture.
Simmer soup over medium heat until broth has thickened slightly, about 3 minutes. Top with green onions.