Cut the burdock into very thin and uniform matchsticks 2-3" long. Cut the carrot and daikon into matchsticks the same length, but a little thicker.
Heat a large cast iron skillet on medium low heat. Add a moderate amount of sesame oil. Add the burdock as soon as a little piece sizzles gently when added to skillet. Add a pinch of sea salt. Sauté the burdock, moving it around constantly. Add a little more sesame oil if necessary. The burdock tends to soak it up.
As soon as the burdock starts looking slightly soft, move it out to the edges of the pan. Add the daikon and a pinch of sea salt and sauté for a minute or two before moving the burdock back in. Sauté the daikon and burdock for a few more minutes. Push those to the edges of pan and continue with the carrot and sea salt. Stir all 3 vegetables together after a minute or two. Sauté for another few minutes.
Pile all of the vegetables in the middle of the pan. Add a few tablespoons of water down the side. Cover, reduce heat slightly and let steam until the vegetables are almost soft. Add a little more water if necessary, but try to keep it minimal.
Remove cover. Sprinkle vegetables with shoyu, stir and let cook until vegetables are done. Remove from heat.
Finely grate ginger, with a microplane zester or ginger grater, and squeeze some of the juice onto the vegetables. Stir it in, taste and add more if desired.
Sprinkle with toasted sesame seeds when serving.