Whisk together the shallots, mayo, sugar, pickle juice, hot sauce, salt and pepper in a bowl. Let it marinate at room temperature for about 15 minutes to help tone down the shallots.
Add half of the cabbage to a food processor and pulse until finely chopped, about ten times. Transfer to the bowl with the dressing and repeat with the remaining cabbage. Mix in the carrots. Season to taste and let it sit in the fridge for at least 2 hours or up to overnight. Adjust the seasoning if necessary.