Combine the mayonnaise, apple cider vinegar, black pepper, garlic, sugar, salt, cayenne pepper, and lemon in a saucepan. Bring to a simmer. Keep warm while cooking the chicken.
Prepare a smoker for indirect cooking over dry heat to 400 degrees F (204 degrees C). For best results, place the coals at least 17 inches (43cm) below the cooking grate.
Rinse the chicken. Remove the backbone by cutting down each side with a pair of kitchen shears. Place the bird breast side-up and press down until the wishbone separates, allowing the chicken to lay flat on the work surface. Pat the skin dry with paper towels.
Spray the bird with olive oil and season both sides with the rub.
Build a nest out of fresh herbs (rosemary, thyme, sage, and chives) on the cooking grate. Reserve a few for placing directly on the hot coals.
Place the chicken, breast side-up, on the herbs and cook until an internal temperature of 165 degrees F (74 degrees C) in the breast and 175 degrees F (79 degrees C) in the thigh.
Remove the chicken from the smoker and rest for 10 minutes before cutting it into pieces.
Drizzle the chicken with White BBQ Sauce and garnish it with fresh herbs. Serve with additional sauce for dipping.