In a bowl, whisk together the mayo, buttermilk, sugar, white vinegar, lemon juice, shallots, 1 teaspoon salt and 1/2 teaspoon pepper.
In 2 batches, add the cabbage to a food processor with the blade attachment. Pulse about 10 times until finely chopped. Pour into the bowl with the dressing, then repeat with the second batch of cabbage. Mix in the carrots and season with additional salt and pepper if needed. Refrigerate for at least 2 hours and up to overnight.
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