Divide the warm cooked rice among 4 deep bowls. Keep warm while you prepare the topping.
In a wide skillet with a lid, combine the dashi, soy sauce, mirin, and sugar. Bring the mixture to a gentle simmer over medium heat, stirring briefly to dissolve the sugar.
Add the sliced onion and simmer for 4 to 5 minutes, until it begins to soften.
Add the chicken in a single layer. Cover and simmer gently for 7 to 9 minutes, turning the pieces once, until the chicken is cooked through and reaches 165°F (74°C) in the thickest pieces.
Reduce the heat to low. Pour the beaten eggs evenly over the chicken and onion. Cover and cook for 2 to 4 minutes, until the egg whites are set and the eggs reach 160°F (71°C). The yolks may remain slightly soft depending on timing.
Spoon the chicken, egg, onion, and sauce over the bowls of rice. Garnish with scallions, if using, and serve right away.