How to Make It
To spatchcock the chicken, place chicken, breast side down, on a cutting board. Using a pair of heavy-duty kitchen shears, cut along both sides of the backbone and remove it. Flip chicken so it's breast side up, then press down to flatten.
Stir hot sauce, paprika, 3 cups buttermilk, 3 teaspoons salt, and 1 teaspoon pepper in a 13-by-9-inch glass baking dish. Add chicken, turn to coat, and place breast side down. Cover; marinate in refrigerator for at least 3 hours and up to 12 hours.
Preheat oven to 400°F. Remove chicken from marinade and pat dry; discard marinade. Place chicken, breast side up, on a large rimmed baking sheet. Drizzle evenly with oil; sprinkle with ¾ teaspoon salt and ½ teaspoon pepper.
Bake chicken until golden brown and a thermometer inserted in thickest portion of thigh registers 165°F, 1 hour, 5 minutes, to 1 hour, 10 minutes. Remove from oven; let rest for 10 minutes. Cut into pieces.
While chicken rests, stir together blue cheese, parsley, shallot, lemon juice, mustard, and remaining ½ cup buttermilk, ¼ teaspoon salt, and pinch of pepper.
Toss beans, sliced celery, and celery leaves on a platter. Serve with chicken and blue cheese dressing for drizzling.