FYI, burnt ends are move of a Kansas City BBQ treat than Austin, Texas. However, if any BBQ place serves brisket and shredded beef there is a good chance they also have burnt ends as well. You should call ahead and ask them to reserve you the brisket tip or burnt ends, as they are usually not a menu item and there is always a limited amount.
With that being said, the best BBQ spots in Austin would be
Rudy's and
Salt Lick.