Scrub burdock root well, then peel roughly to remove the tough outer layer. Julienne into thin strips (about 5cm long). Immediately soak in a bowl of cold water with a splash of vinegar for 5-10 minutes to prevent browning and remove bitterness. Drain well.
Peel the carrot and julienne into similar thin strips as the burdock root.
Heat sesame oil in a large frying pan or wok over medium heat. Add the drained burdock root and stir-fry for 3-5 minutes until slightly softened. Add the julienned carrot and continue to stir-fry for another 2-3 minutes.
Pour in sake, mirin, soy sauce, sugar, and water. Bring the mixture to a simmer, then reduce heat to medium-low.
Cook, stirring occasionally, for 8-10 minutes, or until most of the liquid has evaporated and the vegetables are tender-crisp.
Remove from heat. Stir in toasted white sesame seeds and togarashi (if using). Serve warm or at room temperature as a side dish.
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