In Texas, barbequed meat is usually served with sauce on the side if there is any sauce at all. My favorite restaurant in College Station (I can t remember its name) served half a raw onion, a 4-oz. slice of cheddar cheese, a pickle, and 8 ounces of whatever meat you wanted, all on a piece of butcher paper. They gave you a knife (no fork) and a jar of their own barbeque sauce That would've been Tom's BBQ, and what you describe (the meat, cheese, etc) was the aggie special. Sadly, Tom's went out of business a few years ago after being bought out by someone not skilled in the art.
I was just about to name Tom's for the same quote. Sad to hear they went out of buusiness.
There was a new restaurant that opened on Harvey Road (near the intersection with Texas) that makes some excellent BBQ, though can't remember the name of it. The thought of their sauce is making my mouth water.