In a medium bowl, whisk the eggs; set aside. In a 12-inch skillet, combine the chicken, onion, broth, soy sauce and mirin. Bring to a vigorous simmer over medium-high and cook, stirring occasionally, until the onion is softened, the chicken is cooked through and about ¼ inch of liquid remains in the pan, 8 to 11 minutes.
Pour the egg in an even layer over the chicken mixture. Cover, reduce to medium and cook without stirring until the eggs are just barely set, 2 to 4 minutes. Remove the pan from the heat.
Divide the rice among 4 serving bowls, then spoon on the chicken-egg mixture, dividing it evenly. Sprinkle with pepper and serve with soy sauce on the side.
