Thanks Jim in NC. Given that description, it doesn’t sound that much different from what is available at Smokin’ Ed’s in Smyrna with the exception of the sides. Ed’s shoulder is chopped (not finely), but the meat is well smoked and not via gas-fired anything. Ed’s sides are nothing to write home about but nothing is served on the sandwich unless you ask for it. Ed’s mopping sauce is mostly vinegar and pepper. His serving sauce may have a bit more tomato and sweetness than what you have described, but just ask for it on the side (the meat doesn’t need it). You’ll find Smokin’ Ed’s in a small shack on the right side of Murfreesboro Road as you head south out of Smyrna (just pass Nissan).
Whitt’s may have some similarities to what you’re looking for in that their sauce is mostly vinegar (not sure it has any tomato at all) and their meat is chopped finely, but I’ve never detected any hint of smoke flavor in their meat.