1.Combine water, dashi, soy sauce, mirin, and sugar in a small bowl and mix all together until the sugar is dissolved, set aside. Meanwhile, slice onion lengthwise, about ¼ inch wide and slice green onions on a diagonal.
2.Trim the extra bits of fat and connective tissue from chicken thighs. Cut the chicken thigh along the grain into strips ¾–inch wide. Next, angle your knife back and diagonally, and then slice the chicken strips against the grain into pieces about ¾-inch squares. Transfer the chicken to a bowl and sprinkle with sake. Set aside for at least 5 minutes.
3.Crack eggs into a bowl. Using chopsticks, lift the egg whites to “cut“ them 5–6 times into a few smaller clumps. Do not whisk or beat the eggs together. If the yolks haven‘t broken yet, you can poke them to gently break them, but do not blend the egg whites and yolks. The eggs should resemble a marble pattern.
4.With the stove off, add the sliced onions to a 12" cast iron pan in a single layer then add the seasonings mixture. It should just cover the onions. If not, your pan is too big and you need to either increase the seasoning or use a smaller pan.
5.Turn on the heat to medium and bring it to a simmer. Once simmering, add the chicken on top of the onions. Make sure the onions and chicken are evenly distributed.
6.Once simmering again, lower the heat to medium low. Cook, uncovered, for 5 minutes or until the chicken is no longer pink and the onions are tender. Halfway through, flip the chicken.
7.Now, increase the heat to medium. With the cooking liquid simmering (small bubbles around the edges), drizzle two-thirds of the eggs in a circular pattern over the chicken and onions, avoiding the edges of the pan. If the heat is too strong, lower the heat. If the egg is moving toward the edges of the pan, gently gather it toward the center where the heat is not as strong. Optionally, you can cover the pan if the egg whites are taking a long time to set.
8.When the eggs are still runny but just set, add the rest of the eggs to the center and around the edges of the pan. Add the green onions on top and cook on medium low until the egg is done to your liking.
9.Serve steamed rice in individual serving bowls. Spoon the cooked chicken and egg mixture onto the steamed rice and drizzle the desired amount of pan sauce on top.