Carefully cut the cabbage in half. Remove the core with a knife, then cut the cabbage into pieces small enough to fit into the tube of your food processor. If you are doing this by hand, it is still a good idea to cut the cabbage into manageable pieces.
Grate the cabbage as finely as you can with a food processor, grater, or by hand, then do the same with the carrots.
Combine the carrots and cabbage in a strainer or colander and season them with the salt. Toss to combine. Place the strainer in a bowl or leave it in the sink to drain for an hour.
After about an hour, squeeze the cabbage by hand or by twisting it in a clean dish towel to remove as much moisture as you can.
In a large bowl, combine the sugar and apple cider vinegar. Whisk them together until the sugar has dissolved. Add the mayonnaise (or Vegenaise or Miracle Whip) along with the celery seed, chives, and black pepper. Whisk the dressing again. Taste and adjust the seasoning to your preference.
Squeeze the cabbage mixture one more time. Add it to the dressing and mix until the dressing is evenly distributed throughout the cabbage. Serve.