Pour the oils in a small pot, add the chicken, set the pot over medium heat, and wait for the chicken to sizzle. Cook, stirring occasionally, until the chicken is no longer pink on the outside, about 2 minutes.
Add the carrot, abura-age, burdock and konnyaku and stir well, then add the dashi, soy sauce, sake, mirin and salt. Raise the heat to medium-high; bring the liquid to a strong simmer, and cook until the carrots are tender with a slight bite, 5 to 8 minutes. Use a spoon to skim off any white scum from the surface. Strain the liquid through a sieve into a large heatproof measuring cup, reserving the solids. If necessary, pour off any excess liquid or add enough water or extra dashi to give you 2 cups of liquid.
Combine the liquid and rice in a rice cooker or a medium pot and stir briefly. If you’re using a rice cooker, use the white rice setting if there is one and press the “cook” button. If you’re using a pot, cover it, set it over medium-high heat, and bring the liquid to a boil. Immediately reduce the heat to very low to maintain a bare simmer and cook, doing your best not to peek under the lid, until the rice has absorbed the liquid and is tender, about 15 minutes. If you are using a rice cooker, wait until the rice has finished cooking completely and the timer goes off.
Add the reserved chicken mixture (but don’t stir just yet) and cover with the lid again. Remove from the heat and let the pot or rice cooker sit until the chicken mixture is hot and the rice is completely tender, at least 10 or up to 20 minutes. Stir gently but well, then serve right away.