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 and 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 to the bowl, mixing them with the carrots thoroughly with your clean hands.
Finely chop the green onions mixing, them into the slaw with your hands as well. Add the salt and pepper to the slaw and mix one last time.
In a small mixing bowl combine the remaining ingredients whisking until smooth and creamy. The coleslaw does benefit from being dressed a little in advance to allow for the cabbage to soften a bit.
Once you've mixed the slaw and dressing, transfer it to a serving bowl.