This donburi (Japanese rice bowl) takes some chopping, but once the prep is done it comes together quickly and easily. Just be sure to start about 3 days in advance prepping the tofu so it is ready when you need it. Also, when you make your dashi (stock) to use in making the tofu and vegetable topping, be sure to save the shiitake mushrooms once the dashi is done because you will be using them in the topping!
Drain the tofu and pat it dry with a paper towel. Wrap in a layer of plastic wrap and then a layer of aluminum foil. Freeze in the freezer for 24 hours. Thaw in the refrigerator. Do not unwrap the tofu, but do place it, wrapped, in a container, as it may leak liquid. Thawing may take up to 2 days. When the tofu has thawed completely, unwrap it and shake off the excess liquid, then place it on a large plate lined with several layers of paper towel. Put several more layers of paper towel on top, then place a cutting board on top of that. Weight the cutting board down with something like several cans of food or a book and let the tofu press for at least 30 minutes (preferably several hours). Once the tofu has been pressed, cut it into small cubes and set it aside.
Put the frozen, shelled edamame in a pot and cover them with water. Bring the water to a boil over medium-high heat, then remove the pot from the heat and drain the water. Set the thawed edamame aside.
Scrape off the outside layer of the burdock root using the back of a chef's knife, then cut it. Start at the thicker end and cut it crosswise, with the knife at a 45-degree angle, about 1/2-inch from the end. Keeping the knife pointing in the same 45-degree angle, roll the burdock root toward you, about 1/3 turn, and slice again. Continue cutting this way, adjusting the length and angle of you cuts as needed to produce small chunks. Cut the peeled carrots using the same method. (This is the Japanese rangiri cut.) Cut the peeled Japanese sweet potato in half lengthwise, then cut each half in half again lengthwise. Cut each quarter crosswise into 1/2-inch thick slices.
Combine the dashi, reduced sodium tamari, mirin, and ginger juice in a medium-sized saucepan. Bring to a gentle boil over medium heat. Add the tofu, carrots, burdock root, Japanese sweet potato, and mushroom pieces and simmer, uncovered, until all of the vegetables are tender. Reduce the heat as needed to maintain a gentle simmer as you are cooking the vegetables. Add the edamame and cook for an additional minute, allowing them to reheat and absorb the flavors in the saucepan.
To assemble the rice bowls, put 1 1/2 to 2 cups of hot rice into individual donburi bowls or large deep soup bowls. Cover the rice with the tofu and vegetables. Spoon a few tablespoons of cooking liquid over each bowl. Garnish with thinly sliced green onion and toasted sesame seeds and serve immediately.
If you cannot find burdock root, you can substitute 2 parsnips. To make ginger juice, finely grate several tablespoons of ginger, then put it in several layers of cheesecloth, or a tea bag, and twist to squeeze out the juice. You can also just use freshly grated ginger, but then you will have the fibers from the ginger as well. It's up to you!