A spicy smoked-paprika marinade gives skin-on butterflied chicken a robust flavor. To give the chicken even more flavor, loosen the skin with your hand and spread some of the mixture underneath before grilling.
Combine vinegar, oil, garlic, paprika, and salt in a nonreactive baking dish or large resealable plastic bag.
Remove backbone from chicken using kitchen shears. Place chicken, breast side up, on cutting board, with legs extended on both sides. Press firmly on breastbone to break it and flatten chicken. Add to marinade. Cover dish or seal bag, and refrigerate at least 4 hours (or overnight), turning occasionally.
Preheat grill to medium. (If you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 5 or 6 seconds.) Remove chicken from marinade; discard marinade. Place chicken, skin side down, on grill. Close cover, and grill 15 minutes. Flip chicken, and close cover. Grill until an instant-read thermometer inserted into thickest part of the breast registers 165 degrees, about 20 minutes more.