Just a quick shout out for Southside Market in Elgin. I went there specifically for the mutton BBQ and was not disappointed. (I had traveled there before only to find it closed.) Superb flavor, the mutton, and very unctuous and moist. Fairly priced meats too. The whole operation was quite impressive: clean, and efficient.
I didn't ask them how many people get the mutton, but I bet the cognoscenti keep it turned over. I think there is one place in Austin itself does mutton, but I recall when I went there I found it rather suspect and sorry for itself. This was nice meat. I wish more places did BBQ lamb & mutton, but glad this one does. Oh, and their hot sauce is perfect dressing for mutton: cuts the grease, but does not overwhelm the flavor. Well worth the trip from Eagle Lake (home of Austin's, which can be good, but do cabrito [sometimes] but never mutton/lamb). For BBQ newshounds, there is another place in Eagle Lake (Mike's ) which has been on the cusp of appearing to open for some while. It may have run out of funds. Not sure. Eagle Lake's ex Farris Hotel is also being actively worked on and spruced up now renamed the "Wessex House."