Shred green and purple cabbages into thin ribbons using a sharp knife or mandoline. Grate carrots finely to maintain texture. Toss all vegetables gently in a large bowl to avoid wilting.
In a small bowl, whisk together rice vinegar, soy sauce/tamari, sesame oil, sugar, garlic, and ginger until sugar dissolves. Adjust sweetness/brininess as needed.
Toast sesame seeds in a dry pan over medium heat until golden and aromatic. Add them to the dressing, mixing well.
Drizzle dressing over the slaw and toss to coat evenly. Chill for 10–15 minutes before serving for optimal flavor balance. Serve cold or at room temperature.