Heat chicken broth in a large pot over medium-high heat; stir in green onions.

Whisk 1/3 cup cold water, cornstarch, soy sauce, and sugar together in a bowl until smooth; stir into broth. Bring broth mixture to a boil; simmer for 1 minute, then reduce heat to medium-low.

Whisk eggs and remaining 2 tablespoons cold water together in a small bowl with a fork; drizzle, a little at a time, from the fork into hot broth. Remove from heat and serve.