Make the Basic Barbecue Sauce:

Heat the oil in a saucepan over medium heat and add the onion and garlic. Cook gently, stirring, for about 5 minutes. Add the ketchup, vinegar, Worcestershire sauce, sugar, chili powder, and cayenne to taste (the more cayenne you use, the hotter it will be ). Reduce the heat and simmer, partially covered, until the sauce has thickened slightly, about 20 minutes.

Make Barbecued Chicken:

Prepare a fire for indirect-heat cooking in a covered grill. Indirect-heat grilling cooks food by reflected heat, much like roasting in an oven. This method is good for grilling larger peices of meat, allowing the meat to cook more slowly and evenly, although you may turn the food partway through the cooking time to ensure uniform cooking and distribute appetizing grill marks. See indirect-heat grilling directions below:

(For indirect-heat in a gas grill, first heat the grill using all of the burners, then turn off any burners directly beneath where the food will cook and put a drip pan on the fire grate. Replace the grill rack, put the food over the drip pan, and adjust the burners on either side of the food to equal amounts of heat.)

(For indirect-heat in a charcoal grill, place a drip pan (an aluminum roasting pan is ideal) on the fire grate and use long-handled tongs to position the hot coals around the edge of the pan. Then put the food directly on the grill rack over the pan and cover the grill. For foods that require 40 minutes or more of cooking time, light a second batch of coals in another grill or other fireproof container and use them to replenish the fire as the first batch of coals dies out.)

Sprinkle the chicken pieces on both sides with salt and pepper. Place them, skin side down, around the edge of the grill rack so they are directly above the hot coals. Grill, uncovered, turning frequently, until well browned, 10-15 minutes. Watch the chicken constantly and have a spray bottle of water handy to douse flare-ups. Move the chicken pieces to the middle rack-they may overlap slightly-so that they are no longer directly over the fire.

Cover the grill and open the vents halfway. Cook for 10 minutes. Turn the chicken, brush it with the sauce, re-cover, and cook for 5 minutes more. Brush again with the sauce, then cover and cook until opaque throughout, about 5 minutes longer.

Transfer to warmed individual plates and serve at once. Pass the remaining sauce at the table.