Core the cabbages and slice very finely. This can be done by hand or using a mandolin or a food processor attachment.
¼ red cabbage, ¼ white cabbage
Grate the carrots, apple, and onion. Use the large side of a box grater or the grating attachment in a food processor.
2 medium carrots, 1 large apple, cored, 1 small red onion
Place the vegetables into a large mixing bowl.
Pour the vinaigrette over the salad and use your hands to mix until well combined. Serve immediately or, if you have the time, allow it to rest for at least 30 minutes first.
Vegan coleslaw will keep for several days in a sealed container in the refrigerator. It can be eaten cold but is best served at room temperature.
Vinaigrette
Add the olive oil, vinegar, mustard, salt, and pepper to a small jar or bowl. Shake or whisk until well mixed and the vinaigrette is fully emulsified.
3 tablespoons olive oil, 2 to 3 tablespoons apple cider vinegar, 2 teaspoons Dijon mustard, ½ teaspoon sea salt, ¼ teaspoon black pepper
Notes
Vegan coleslaw lasts for several days if refrigerated. It's essentially pickled, so the vinegar and salt preserve the vegetables. Wilting isn't an issue here and the salad tastes even better after a day or two, though it's best served at room temperature.