Picking the Pieces: Chicken is really two kinds or meat. The white meat is very different in texture and flavor from the dark meat. For this reason you can choose the kind of meat you want to make your chicken barbecue. Of course, by using whole chickens you will get a variety in one great pile of shredded meat. If you choose chicken breasts, be careful not to let it dry out. While you will be slathering the meat with a delicious barbecue sauce, dry meat will be tough and chewy, not something you want in a barbecue sandwich. I prefer smoking whole chickens because I like having the different meats in one sandwich. Though, it is hard to resist just doing my favorite piece, the thigh. While the white meat is low in fat and has an even texture, the dark meat is much more flavorful.
Cooking the Chicken: The secret of good barbecue is low and slow. If you were preparing a big pork shoulder you'd be looking at a minimum of 12 hours of cooking time. Chicken on the other hand, can be cooked at a hotter temperature in a shorter amount of time. This means you can either use your smoker at a temperature between 225 and 250 degrees F. If using your grill, cook indirectly with a low flame. The whole goal of this cooking process is to cook the chicken, allowing fat to drain from the meat, while keeping it tender and moist. Remember, you can't overcook chicken, but you can dry it out. Chicken must reach an internal temperature between 175-180 degrees F., before it is done. Do not remove the chicken from the grill or smoker until it reaches this temperature or you could be sorry, very sorry.
Adding the Smoke: All great barbecue needs smoke. Smoke not only adds that authentic barbecue flavor but the nitrates in the smoke break down the connective tissues and changes the very nature of the meat. If you are using a smoker you should already know how to get the smoke. If grilling indirectly, you will need to make smoke. Of course, you wil be adding barbecue sauce later, but the flavor just isn't the same. More smoke during the cooking means less flavor you have to add later. You don't want to overpower the chicken with a thick store bought sauce.
Shredding the Chicken: When the chicken is completely cooked, let it sit for about 15 minutes to even out the heat and allow it to cool down enough to handle. While getting the chicken ready to serve, you will need to reheat it, so it doesn't matter if it cools down a lot. The big question about shredding is how to do it. Traditionally, chicken is pulled apart by hand. This is time consuming so you might want to consider chopping or pulsing in your food processor. There are advantages to each. Hand pulled chicken is more authentic and it gives you the ability to really get into the meat. Chopping is easier but tends to crush the meat as you chop. Pulsing in the food processor is quick and easy but can result in ground chicken that really isn't very appetizing. However, the more shredded the meat the more sauce it will hold, so you might want to consider a combination of two of these methods. Personally, I tend to chop up the breasts because they are large pieces of meat and can be quickly dealt with by using a sharp knife. I use my hands to pull apart the dark meat because the bones make it harder to carve and this way I can make sure to get rid of any leftover fat deposits. It is important to get a feel for the cooked chicken to decide which method works best for you.
Adding the Sauce: Most people will use a sweet tomato-based barbecue sauce on their pulled chicken barbecue. This works well because the sweetness of the sauce adds to the flavor of the chicken. If you have added smoke to the chicken during cooking, then avoid using a smoky sauce. You can go with more traditional sauces that use vinegars and savory flavorings. The choice is entirely up to you. Use enough sauce to lightly coat the chicken. You don't want the shredded chicken swimming in sauce. The meat should be evenly and completely coated, no more.
Serving your Chicken Barbecue: Nothing fancy please, this is barbecue after all. Pulled chicken should be piled high on a bun that does nothing more than give you the ability to hold the sandwich. It is a means to a sandwich, not a gourmet experience. If you would prefer, top off your pulled chicken sandwiches with cole slaw and set a pickle spear next to it. Now you have a great sandwich.