To make the sauce, in a small frying pan over medium-high heat, combine sake, soy sauce, mirin, sugar and broth and bring to a simmer.
Add chicken to pan and simmer until chicken is half cooked, about 5 minutes. Add onion, and cook until chicken is cooked through and onion is soft, about 5 minutes longer.
Meanwhile, break eggs into a bowl and beat with a fork or chopsticks until well- blended. Divide rice into two wide, shallow bowls.
When chicken is ready, add ¾ of beaten egg to pan, cover, and cook until egg has just set, 4–5 minutes. Uncover, pour in rest of egg, and immediately pour chicken mixture over bowls of rice, dividing it evenly. Garnish with green onions, if using, and serve right away with chopsticks and spoons to get every last bit of rice and sauce.