Preparing the vegetables. Wash the burdock root and carrot under running water to remove any soil. Use a vegetable or clean kitchen cloth, if necessary, to get the skin perfectly clean, since the vegetables will be prepared unpeeled. Cut both vegetables into 2.5" lengths that are 1/8" thick. You can julienne them, or shave them at an angle. The important thing is to get the thickness right so that it cooks quickly and evenly.
2 Cooking. Heat the oil over medium heat, then add the burdock root. Add in a few tablespoons of dashi (if you have it) or water as needed to help steam it. Once the burdock is tender, stir in the carrot and cook for 1 minute. Add the sugar and chili flakes. Continue cooking until the carrots are tender.
3 Seasoning. Pour in the mirin, sake (if using), and soy sauce, and increase the heat to high. Cook until the liquid in the pan is reduced to approximately 1-2 tablespoons. Remove the pot from the heat and add the sesame seeds. Stir to combine, and allow to cool to room temperature before serving.
Note
CUTTING THE VEGETABLES. A sharp knife works well, but if you find it easier, you can use a Japanese vegetable slicer called a benriner to cut the vegetables quickly and uniformly. If you do use one, please be careful and use the included finger guard. The blades are extremely sharp.
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