Cut the burdock root into thin matchsticks or rounds and soak in water for 10 minutes to prevent discoloration. Drain well.
Heat sesame oil in a pan over medium heat. Add burdock root (and carrot if using) and sauté for 2-3 minutes.
Add soy sauce, sake, mirin, and sugar. Stir well to coat the vegetables evenly.
Reduce heat to low and simmer for 5 minutes, allowing the flavors to absorb.
Remove from heat and sprinkle with sesame seeds and red pepper flakes if desired.
Serve warm or at room temperature as a side dish or with rice.