Reserve 3/4 cup of chicken broth, and pour the rest into a large saucepan. Stir in chives, salt, and ginger; bring to a rolling boil. Stir reserved 3/4 cup of broth and cornstarch until smooth. Set aside.
Whisk eggs and egg yolk together in a small bowl with a fork. Using a fork, drizzle eggs, a little at a time, into boiling broth. Eggs will cook immediately.
Stir in cornstarch mixture gradually until soup reaches desired consistency.