Heat oven to 400 degrees. Line a baking sheet with a wire rack. Combine buttermilk, mustard, salt, and cayenne in a nonreactive dish. Dip chicken in mixture and transfer to dish with breadcrumbs, pressing to coat. Transfer to baking sheet. Drizzle with oil. Bake until golden and chicken is cooked through, about 16 minutes. Season with salt.
Meanwhile, combine buttermilk, yogurt, chives, and salt in a bowl for dressing. (Add 1 Tbsp. crumbled blue cheese, if desired.)
Serve chicken with dressing, celery, romaine hearts, and hot sauce.