Heat oven to 375° F. Set a wire rack inside a rimmed baking sheet and lightly coat with oil. In a shallow baking dish, combine the paprika, 1 cup of the buttermilk, and ½ teaspoon each salt and pepper. Add the chicken and turn to coat.
Place the crushed crackers in a shallow bowl. Remove the chicken from the buttermilk mixture and coat with the crackers, pressing gently to help them adhere. Place the chicken on the rack and bake, turning halfway through, until crispy and an instant-read thermometer registers 165° F in the center of the thickest piece of chicken, 40 to 45 minutes.
Meanwhile, in a large bowl, whisk together the mayonnaise, vinegar, the remaining ¼ cup of buttermilk, ½ teaspoon salt, and ¼ teaspoon pepper. Add the broccoli or broccoli slaw, carrot, and shallot and toss to combine. Serve with the chicken.