Scrub the burdock root and cut into thin matchsticks. Cut carrot to the same size. Mix soy sauce, mirin, sake, and sugar in a small bowl.
Soak burdock in cold water for 5 minutes to remove bitterness. Drain well.
Heat sesame oil in a frying pan over medium-high heat. Stir-fry burdock and carrot for 3–4 minutes.
Add the seasoning mixture and reduce heat to medium. Cook, stirring occasionally, until the liquid is almost fully absorbed — about 5–7 minutes.
Transfer to a bowl and top with white sesame seeds. Serve warm or at room temperature.
