For the sauce: Heat the oil in a medium, heavy-bottomed saucepan over medium heat. Add the garlic and cook, stirring, until fragrant, about 1 minute.
Stir in the ketchup, brown sugar, mustard, Worcestershire sauce, vinegar, hot pepper sauce, lemon juice and cayenne pepper. Bring to a simmer, stirring. Reduce the heat to very low and simmer, stirring often to avoid scorching, for 15 minutes. Remove the sauce from the heat and stir in the liquid smoke. Use immediately or cool completely, cover and refrigerate for up to 5 days.
Preheat the oven to 375 degrees F.
For oven-baked chicken: Rinse the chicken under cold water and pat dry with paper towels. Arrange the chicken in a single layer in a large roasting pan. Brush the chicken with the oil and season with salt and pepper to taste. Cover tightly with aluminum foil and bake for 45 minutes.
Skim the fat from the pan juices. Spread all of the sauce over the chicken, return the pan to the oven and bake, uncovered, until the chicken is glazed and shows no sign of pink when pierced at the bone, about 15 minutes.
Transfer the chicken to a large platter. Stir the sauce and juices that have accumulated in the pan and pour them over the chicken. Serve hot.
For grilled chicken: Preheat the oven to 375 degrees F. Bake the chicken for 45 minutes in the preheated oven.
In the meantime, build a charcoal fire in an outdoor grill and let the coals burn down until covered with white ash and medium-hot (you should be able to hold your hand over the coals for about 3 seconds). If using a gas grill, preheat on high, then adjust the temperature to medium. Remove the chicken from the oven, transfer it to the grill and cook, basting often with the barbecue sauce, until glazed, about 15 minutes.