Pour the salt into a medium-sized bowl of cold water. Add the potatoes, lightly scrubbing them under the water with your hands to get rid of excess starch, then drain and rinse them in more cold water. If they are a bit large, cut them into chunks.
Place the potatoes in a saucepan, add enough water to cover, then add the vinegar. Bring to a boil and cook until done, about 8 to 10 minutes. Check for doneness. Drain the potatoes and lightly rinse them in cold water.
In another saucepan, boil the mirin for one minute. Add the potatoes and dashi stock, bring to a boil again, and add the soy sauce and sugar. Partially cover the pan, and cook until 80 percent of the stock has boiled off. Shake the pan occasionally to help color the potatoes with the sauce. Remove the potatoes from the heat, drizzle with leftover sauce, top with ginger, and serve at room temperature.
Try making the same recipe with sweet potatoes if that is all you have on hand or prefer to white potatoes.