Beat eggs until the whites and yolks are well combined. Set aside.
In a large pot, bring chicken stock to a gentle boil over medium high heat. In a small bowl, combine the cornstarch and water to make a slurry. Slowly add the the cornstarch slurry into the chicken stock while constantly stirring. Make sure to add slowly and to stir constantly to avoid lumps. If lumps form, you can pass the soup through a strainer.
Once the soup thickens and simmer, add the sesame oil and stir. Slowly pour the egg into the soup in a steady, circular motion, forming egg strands in the soup. Let egg cook for about 15 seconds, and stir the soup in a circular motion.
Add salt and pepper to taste, and add the green onions and stir well. To serve, add soup in a bowl and top with ginger and some more green onions.