Not many know this, but Kentucky has its own style of BBQ, and it revolves around mutton or sheep. The mutton is slow-cooked for around 12 hours and then served with a vinegar-based BBQ sauce. One of the best places to try Kentucky-style BBQ is at Moonlite Bar-B-Q Inn found in Owensboro. While you're there, also check out their assortment of homemade pies, which are a great dessert to finish off your meal.
This is another really old BBQ joint, and they've been serving up barbecued mutton since the 1950s, so you know you're in good hands when you give this place a visit.