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 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 mayo, mustard, sugar, vinegar, celery seeds, garlic salt, and pepper. Pour enough of the dressing over the grated vegetables to moisten them.