Thinly slice the both the green and the red cabbage and place in a large bowl.
Optional: Purge the cabbage. Combine the cabbage in the bowl with 1 tablespoon of kosher salt and allow it to sit for ten minutes to purge some of its water and soften the cabbage. Rinse well in a colander. Dunk the cabbage in ice water to crisp it up, drain, and spin in a salad spinner to remove excess moisture.
Shred the carrot on a box grater or mandolin slicer with the medium toothed insert. Cut the onion in half and thinly slice into half rounds, separating the pieces.
Combine the cabbage, carrot and onion in a large bowl.
In a small bowl, whisk together the mayonnaise, vinegar, sugar, celery seed, salt and pepper until smooth.
Slowly add the dressing to the cabbage mixture until you have the desired about of dressing. If you have purged the cabbage you may find you need less dressing. With completely raw fresh cabbage, you may need all of it. Taste for seasoning and adjust as needed.
Options: Add red onion, thinly sliced green apple, celery or fennel to the slaw mix. In the dressing, add mustard, horseradish, or hot sauce for extra flavor.