Cut the chicken into bite sized pieces. Slice the onion into 1cm wide slices and the leek into 7 mm wide diagonal slices. Slice the ginger thinly.
Heat some oil in a frying pan and stir fry the ginger and chicken, until the chicken is browned on the surface.
Add the onion and leek and stir fry until browned. Add the rest of the ingredients and bring to a boil. Skim off any scum.
While this simmers, beat the egg lightly. I recommend beating it just enough so that the yolk and white aren't totally mixed together.
When the sauce in the pan has reduced by a bit less than half, swirl in the beaten egg and put a lid on. When the egg is soft-set (or to the doneness you like) it's done.
Serve on top of rice.
It looks nicer if you scoop it on top of the rice carefully with a spatula so that you don't mess up the soft-set egg.
I edited the amount of honey needed, since I think some people like it sweet while others might not. Personally I like to use 2 tablespoons, since I prefer this on the sweet side.
Story Behind this Recipe
I've been making oyako don this way forever. By stir-frying the ingredients, I think it comes out tasting a lot richer.
Brown the chicken thigh meat, onion and Japanese leek on on surface. They'll be simmered later so there's no need to cook them through at that point. If you don't have any honey, use sugar or sugar syrup instead. Try not to beat the egg too much. I think it tastes creamier that way. If you don't like ginger, you can just take it out during the cooking process.