Finely chop the cabbage and carrot using a food processor. Combine them in a large mixing bowl and set aside.
In a small mixing bowl, whisk together the buttermilk, white vinegar, milk, mayonnaise, lemon juice, sugar, salt, and black pepper until the mixture is smooth.
Pour the prepared dressing over the cabbage and carrot mixture. Stir thoroughly to ensure the vegetables are evenly coated.
Transfer the coleslaw to the refrigerator and let it chill for at least 30 minutes before serving.
Notes
For the best flavor, let the coleslaw sit in the refrigerator for 30 minutes.
Use fresh ingredients for better texture and flavor.
Adjust seasoning to taste and add more milk if dressing is too thick.