The one truth about barbecue seems to be that people use whatever meat they have on hand. In Texas it's beef, in the Carolinas it's pork, and in Western Kentucky it's mutton. Thanks to the tariff of 1816, wool production in the then Western United States, became profitable and suddenly people found themselves with a larger population of sheep.
Any story of the origin of barbecue starts with a meat that is too tough and undesirable to be sold for a profit. Mutton barbecue is no different. Aging sheep who no longer produced good wool became a virtually unlimited resource, but the meat was too tough and too strong tasting to be worth anything, so people turned to the tried and true methods of low and slow cooking. In the early days, a whole sheep would be cooked for long hours over a low fire. A mixture of salt water would be mopped over it and it would be served up with a dipping sauce of vinegar and hot peppers and stuck between a couple slices of bread. In Kentucky, this "sauce" is called a dip, specifically Mutton Dip or Vinegar Dip.
In more modern times, people have put aside the whole sheep for select parts, particularly shoulder roasts which are similar in many ways to the pork shoulder roasts used in Carolina Barbecue.
Today the best place to get this kind of barbecue is the Old Hickory Pit Bar-B-Q in Owensboro. Here, large pieces of mutton are cooked for 12 hours at temperatures in the high 200's while being mopped with a mixture of water, Worcestershire sauce, vinegar, lemon juice, salt, and pepper.
As for the method, it's a pretty basic barbecue procedure. Get your smoker ready for a long smoke at about 220 degrees F. (105 degrees C.). Place the prepared piece of mutton (or lamb) fat side up in the smoker. Plan on smoking for about 1 1/2 hours per pound. Mop every hour and remove when the meat is around 170 degrees internally. Mutton barbecue is typically served sliced on buns with a table sauce. See the recipes on the top right-hand side of this page for specific suggestions.
Sadly, mutton barbecue is becoming something of a lost art. The Moonlite Bar-B-Q Inn says that they now serve as much pork as they do lamb and there are few people stepping up to the plate to continue the traditions of this unique style of barbecue so if you try it and like it, pass it on.