Cut cabbage in quarters and remove 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-holed grater attachment and push cabbage, carrots, and onions through feed tube to grate. In a large bowl, toss vegetables together.
In another medium bowl, prepare the coleslaw dressing by whisking together the mayonnaise, mustard, cider vinegar, sugar, black pepper, and cayenne. Toss dressing with the cabbage mixture, and season with salt and pepper, to taste. Cover with plastic wrap and chill for at least 2 hours before serving.