Cut cabbage in quarters and remove the core. Peel carrots and onion and cut into pieces that would fit through the feed tube of a food processor. Fit food processor with the large grater attachment and push cabbage, carrots, and onions through feed tube to grate; toss the vegetables together.
In another medium bowl, prepare the dressing by whisking together the mayonnaise, mustard, cider vinegar, sugar, black pepper, and cayenne. Continue mixing this together until sugar is somewhat dissolved. Toss dressing into the cabbage mixture, and season with salt and pepper, to taste. Cover with plastic wrap and chill for at least 2 hours before serving.
Dijon mustard can also be substituted for the yellow mustard.