Bring a large pot of water to a boil. Cook soba noodles in boiling water until tender, about 5 minutes. Drain noodles in a colander, rinse with cold water, and set aside.

Pour rice vinegar, soy sauce, lime juice, and sesame oil into a large bowl. Mix in brown sugar, lime zest, garlic, and red pepper flakes; stir until sugar dissolves. Toss in carrot, cilantro, and peanuts until well combined.

Cut noodles into 3-inch lengths. Stir into carrot mixture. Cover the bowl and refrigerate for at least 1 hour.