Cut the cabbages into quarters, and remove the cores. Peel the carrots and onion, and slice them into pieces that will fit through the feed tube of a food processor. Fit the food processor with the large holed grater attachment, and push the cabbage, carrots, and onion through the feed tube to grate. Pour the vegetables into a large bowl, and toss to combine.
Whisk together the mayonnaise, mustard, vinegar, sugar, black pepper, and cayenne in a medium bowl, whisking until the sugar is dissolved. Toss the coleslaw with the dressing, and season with salt and additional pepper to taste. Cover the slaw with plastic wrap, and chill for at least 2 hours before serving. Enjoy!