Make the sauce. Put the mirin and sake in a pot and bring to a boil.
Add the soy sauce and sugar, and bring to the boil again.
Cut the vegetables and chicken. Cut the onion in half, then slice lengthwise into about 5 mm thick. Cut the chicken into bite sized pieces. Chop off the root ends of the shimeji mushrooms and shred.
Take out about half of the sauce, add the chicken to the remaining sauce and add water to cover.
When the cooking liquid comes to the boil and the chicken is just about cooked through, take it out. Skim off the scum from the liquid.
Add the onion to the cooking liquid and simmer until transparent, then add the shimeji mushrooms. The topping is done. Taste and adjust the seasoning if needed by adding some of the reserved sauce.
Transfer to another pan to cook the egg, add the chicken pieces, pour in the beaten egg and cover with a lid. Cook until the egg is soft-set over low-medium heat. In our house, we make 4 servings at a time in a frying pan.
Put some rice into bowls and pour the egg-chicken mixture on top. Add the leftover simmering liquid, sprinkle with nori seaweed and it's done.
Story Behind this Recipe
My father didn't like chicken, so I never had a proper oyako don until I was grown up. I used the type of oyako don you get at a soba noodle restaurant as a reference.
It's important not to simmer the chicken for too long. It will continue to cook in residual heat, so check to see if it bounces back when you press it with chopsticks. Beat the eggs well before adding it, otherwise the eggs will be two-toned. Store any leftover sauce in the refrigerator. It will keep for about 2 months, and can be used for teriyaki or simmered dishes.