Fit a food processor with the thickest slicing blade. Cut the cabbages into small wedges and place horizontally into the feed tube. Process in batches. Next, fit the food processor with the grating blade. Cut the carrots in half and place in the feed tube so they are lying on their sides. Process in batches and mix in a bowl with the grated cabbages.
In a medium bowl, whisk together the mayonnaise, mustard, sugar, vinegar, celery seeds, celery salt, salt, and pepper. Pour enough of the dressing over the grated vegetables to moisten them. Serve cold or at room temperature.
Placing the vegetables horizontally in the food processor ensures long shreds.
The vegetables can be grated a day ahead and stored, wrapped in plastic, in the refrigerator. Keep the red cabbage separate, or it will turn the other vegetables pink.
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