Remove fat and skin from chicken, cut into small pieces, sprinkle sake onto the chicken and let stand for about 10 minutes.
Cut the carrot into julienne strips (very thin!).
If using dried shiitake, soak in 1/2 cup of water to soften, remove stem and cut into julienne strips. If using regular button mushrooms, just go ahead and cut into little pieces skipping the soaking step (same if you are using other type of fresh (not dried) mushroom).
Pour the stock, mushroom soaking water (or water and extra stock), chicken, carrot, and mushroom into a pot and heat.
When it begins to boil, add sugar and sake, and simmer for about 2 minutes Add soy sauce and sale, and continue to simmer for a couple more minutes.
Remove the solids from the pot (use a colander with a pot/bowl underneath if you need to), and let the liquid stand until cool (if using a rice cooker, liquid should be cool; if stovetop, then it doesn't matter).
Place the rice into the liquid. If there it is a bit dry, add more warm water. Add the solid ingredients back in, and cook (or transfer to rice cooker and cook as usual).
When it's done, keep covered for about 10 minutes. Mix the rice so that the ingredients will spread evenly.