Bring first 4 ingredients to a boil in a Dutch oven. Reduce heat, and simmer, uncovered, 10 minutes. Add tofu and next 3 ingredients. Bring to a boil; reduce heat, and simmer, uncovered, 5 minutes.
Combine water and cornstarch in a small bowl; stir well with a whisk. Add cornstarch mixture to soup. Bring to a boil, and cook 1 minute or until soup begins to thicken. Reduce heat to medium.
Pour egg in a slow, steady stream into simmering soup. Stir in sesame oil. Garnish with green onions, if desired. Serve immediately.