In a large saucepan, combine 3¼ cups chicken broth, the chives, salt, and ginger and bring to a rolling boil.
In a small bowl, stir together remaining ¾ cup broth and the cornstarch and mix until smooth. Set aside.
In another small bowl, whisk the eggs and egg yolk together using a fork.
Drizzle egg mixture a little at a time from the fork into the boiling broth mixture (egg should cook immediately).
Once all the eggs have been dropped, stir in the cornstarch mixture gradually until the soup is the desired consistency. Top with fried wonton strips and serve warm.