Combine the dashi, soy sauce, mirin, and sugar in a wide frying pan. Bring to a simmer over medium heat. Add the sliced onion and cook for 3–4 minutes until softened.
Add the chicken pieces in a single layer. Simmer over medium heat for 5–6 minutes, turning occasionally, until the chicken is cooked through.
Beat the eggs lightly — leave some streaks of yolk and white. Pour about two-thirds of the egg over the chicken and onion. Cover with a lid and cook on medium-low for 1 minute. Pour the remaining egg over the top, cover, and cook for another 30 seconds. Turn off the heat while the egg is still slightly underdone.
Add hot rice to each bowl. Top with a layer of shredded nori. Carefully slide the chicken and egg mixture over the top. Finish with sliced green onion. Serve immediately.