Hard boil 12 eggs for 11 minutes. Peel and chill. If you prefer to hard boil your eggs but still want that smoke flavor, start your Grill on 'smoke' with the lid open until a fire is established in the burn pot (3-7 minutes). Preheat to 225F and smoke the eggs until golden brown, 10 to 15 minutes. Take the yolks of 12 hard-boiled smoked eggs, with the halved whites reserved, and blend together with the mayonnaise, vinegar, mustard, salt, and pepper, and mix well. Place the filling evenly among the egg-white halves. Sprinkle each one with a pinch of paprika and it’s ready to serve.