Finely chop the cabbage and carrot into very small pieces, roughly rice-sized. Finely mince the onion. A food processor can help achieve the fine texture, but a sharp knife works well too.
In a medium bowl, whisk together the mayonnaise, buttermilk, milk, sugar, lemon juice, and white vinegar until the dressing is smooth and well combined.
Add the salt and black pepper and whisk again.
Place the chopped cabbage, carrot, and onion in a large bowl.
Pour the dressing over the vegetables and toss thoroughly until everything is evenly coated.
Cover the bowl and refrigerate for at least 2 hours, or up to 8 hours, before serving. This gives the flavors time to blend and allows the dressing to soak into the finely chopped vegetables.
Stir the coleslaw before serving and adjust the seasoning if needed.