Sam's BBQ on E. 12th is a treat and a short cab ride. Brisket, sausage and pork ribs are hard to beat. The BBQ sauce is a tad sweet for my taste, so I always order it on the side. It seems like every movie or TV show shot in Austin features a glimpse of Sam's. There's also a great short video called Sam's BBQ on YouTube that captures the essence of the place. It's also open after-hours, till 3 or 4am. I think the NYC folks will have something to talk about for a while.

I love Sam's. I've eaten there for over 20 years, though I don't get over as often as I did when I worked downtown. The brisket is indeed melt-in-your-mouth sublime. I also like their pork ribs and sausage. And I agree with you totally on the mutton. I've never seen it on any other BBQ menu!
Lot's of history at Sam's: check out the autographed photos on the wall that survived the big fire in the early 90's. It's a popular after-hours spot, open till 4:00am, or, it used to be. There's a great short video on YouTube that captures the soul of the place. Just search Sam's BBQ.