The term "barbecue chicken," is somewhat of a misnomer, as barbecued chicken does not follow any of the normal conventions that, say, barbecue ribs or brisket do.
The confusion arises from two different definitions of the word "slow," and two different definitions of the word "barbecue."
Southern barbecue is a method of "slow" cooking, but it doesn't simply mean that the cooking takes a long time. It does, but there's more to slow cooking (definition one) than that. At the molecular level, it's a process similar to say, braising or stewing. Meats high in connective tissue are cooked until that connective tissue breaks down into rich gelatin, causing the meat to tenderize. Southern barbecue is, by definition, cooked to a degree well beyond well done. It's only because of the magical lubricating powers of gelatin that the meat still tastes moist and succulent. That's what's meant by slow cooking.
"No waiting around for connective tissues to break down, no gelatin to help things along."
Meats low in connective tissue—say a beef tenderloin—make poor barbecue (or braising) choices, because once they're overcooked, they have no gelatin to rescue them from dryness. They benefit much more from "fast" cooking—methods in which you bring it up to a final temperature, and serve it pretty much straight away. No waiting around for connective tissues to break down, no gelatin to help things along. Think grilled steaks, roasted chicken, or seared duck breasts.
The confusing part is that even fast cooking methods can be done very slowly. A chicken roasted in a 150°F oven, for instance, may take 5 hours to get to its final serving temperature, but because very limited connective tissue breakdown is taking place, it's still technically a fast cooking method.
Confusing, right? Why am I telling you this? No reason other than I think it's interesting, and that barbecue chicken is a perfect example of slow-cooking fast cooked food.
Barbecue chicken doesn't fall under the strict definition of the Southern term "barbecue," as it is not cooked hot or long enough for connective tissue to break down the way it does in ribs or a pork butt (indeed, there isn't really any connective tissue to break down in the first place), but it does fall under the wider umbrella of "barbecue" which includes any foods cooked slowly (not to be confused with slow-cooked) with the addition of smoke and a barbecue sauce.
So how do we not-barbecue-but-barbecue chicken? The key is to start with a very large bird. You want a bird big enough that it can sit in the smoky environment of the closed grill for a good chunk of time to absorb flavor. A 6 to 8 pound roaster is what you're going for. It'll take a good 45 minutes to an hour to cook, start to finish, plenty of time to absorb smoke from the couple of soaked wood chunks you throw on the fire.
Using a two-level indirect fire is the best way to ensure gently cooking without burning the exterior of the bird. After giving them a rubdown with a simple rub of sugar, salt, and a few spices, I start my chickens skin-side-up on the cooler side of the grill with the legs pointed towards the hotter side and cook it covered until it comes up to around 120°F.
From there it gets a few coats of barbecue sauce. You can use your favorite bottled sauce or homemade recipe, but I used Josh's Kansas City-Style Sauce for this bird. The key is to apply sauce in layers, painting it on, covering and cooking, and letting it dry out to tacky stage before applying the next coat. In this way, you build up a nice lacquered layer that develops some really nice caramelized notes.
I flip the chicken over and cook directly over the hot side of the grill just for the last few minutes to crisp and char the skin. The result is a richly seasoned, sweet-and-tangy chicken with deep smokiness and ultra-tender and juicy meat.
All conversation of whether or not it's proper to call it barbecue will end once you all agree that it's delicious.
For more on the basics of grilled chicken, check out The Food Lab: How To Grill A Whole Chicken »