Cut the red and green cabbages in half, from the top through the core, and then cut those halves in half again in the same direction.
Remove the remaining core, then cut very thin slices 90 degrees to the layers, called a chiffonade, using two of the green cabbage quarters and one of the purple. Add to a large mixing bowl, and put the remaining cabbage in the fridge in a plastic zipper bag.
Shred the carrots on the coarse side of a box grater and add them to the bowl with the cabbage.
Finely chop the green onions and mix them into the slaw. Add the salt and pepper to the slaw and mix one last time.
In a small mixing bowl, whisk together the dressing ingredients until smooth and creamy.
Combine the slaw and dressing, then refrigerate for at least 30-45 minutes before serving. The coleslaw benefits from being dressed a little in advance, allowing the cabbage to soften a bit.