In my part of the world, “barbeque” doesn’t mean “chicken”. Like, if you went to a barbeque and they were serving chicken, people would ask “I thought you said that this was a barbeque”.
Nothing wrong with grilling chicken, of course. I’m a fan of marinating the thighs in spicy mustard overnight and then grilling them the next day.
I don’t call it “barbeque”, though. Not here. Wouldn’t be neighborly.
As such, barbeque refers mostly to pork shoulder but, sometimes, to beef brisket. It’s got a bit of a dry rub on it but the expectation is that you slather it in sauce once it comes to you and it is most likely to be the case that the sauce is vinegar-based rather than tomato-based or mustard-based.
“Mixing it up” consists of serving the above on a baked potato rather than as a sandwich. If you’re mixing it up, beans are appropriate. If you are not, cole slaw is appropriate. Okra as a second side.
I have heard that there are places that do this sort of thing with mutton. I would find myself pleased but confused, as “I thought that they said something about this being barbeque.”
Perhaps they do something different where you live and they call it “barbeque” there.
So… whad’ya got?