Yeah, I thought of Moonlite as well. Decent place, but I’m not a fan of the buffet style myself. You also have Old Hickory there in Owensboro serving mutton, as well as Ole South. There was a fourth one there, too, George’s, that was awesome, but closed down a few years ago, sadly. Moonlite was my least favorite of the bunch, but it’s still very good 'cue. Also, if you head across the river directly north to Evansville, IN, you might find mutton there. Last time I was in that area a few years ago, I did see it at some place, but I forget what it was called. But that Western Kentucky area (and just across the border into Indiana) is, as far as I know, the only area in the US where BBQ mutton is the meat of choice. I basically have five barbecue meccas: the Carolinas, Texas, Kansas City, Memphis, and Owensboro.
The only bbq buffet I’ve been to is Bullock’s in Durham, NC. No idea if it’s still open. Every place I’ve been to in the DC area is counter service.
Hehehe, not a bad set of BBQ meccas, at all. Thanks for the heads up on the other mutton outlets in the area! I’ll see if i can draw the relatives to one of them next time we are around there.
Pretty much ALL of them in Texas are the order the meat at the counter, then have buffet-style side dishes, and a very few are order at the counter, and pick it up when they call your name. It’s an outgrowth of where the barbecue developed; in Central Texas, it was a meat market thing, so you went to the meat counter and bought the meat by the lb, and went and got sides elsewhere. Most of them still allow you to buy the various meats by the lb and just combine on your plate as well.
Can’t say I’ve ever been to a sit down barbecue place with menus and waitresses. I think I would be very suspicious of a barbecue place like that, as a rule of thumb for barbecue joints is that the more deeply ramshackle the place seems to be, the better the barbecue will be.
Around DC, mostly sit-down table service. I know of at least a couple of cafeteria-style joints in the DC 'burbs. A lot of counter-service-and-free-seating places around Baltimore. So it mostly depends on where exactly you go.
Our two favorites in DC and close by are Federalist Pig and Sligo Pit, both of which are counter service. Urban BBQ is a local chain that we often hit after church (sometimes the majority of customers when we go are members of our church) is also counter service.
The one dedicated BBQ joint we’ve found in our rural county is a Dickeys order-at-counter franchise whose quality dropped with new owners so meh. A few regular sit-down eateries here serve alleged BBQ but not even as weak as Dickeys. A couple of BBQ trucks supposedly are based nearby but we’ve never seen them, only taco wagons. We’re in a BBQ desert. No good Indian or Chinese buffets, either. Beyond the fucking edge of civilization, here. 
But there’s a killer handmade ice creamery just down the road. That assuages many pains.