Heat chicken broth and green onions in a large pot over medium-high heat. Mix sugar, soy sauce, cornstarch, and 1/3 cup of cold water in a bowl until smooth. Reduce heat and stir soy sauce mixture in to the broth. Boil soup on high for 1 minute, then reduce heat to medium-low.
Beat 3 eggs with 2 tablespoons of cold water. Using a fork, stir egg mixture in to soup and cook until eggs are opaque, about 2 minutes.