In 3 1/2- to 4-quart slow cooker, place beef. In 4-cup measuring cup or bowl, mix 1 1/2 cups of the barbecue sauce and the root beer; pour over beef.
Cover; cook on Low heat setting 10 to 12 hours.
About 20 minutes before serving, remove beef from slow cooker; place on large plate. Pour juices from slow cooker into 12-inch skillet. Cook over medium-high heat about 15 minutes, stirring occasionally, until juices are thickened and reduced to about 3 cups. Meanwhile, shred beef with 2 forks; return to slow cooker.
Stir remaining 1/2 cup barbecue sauce into reduced juices in skillet; pour over shredded beef in slow cooker. Stir in salt and pepper to taste. Spoon about 1/2 cup beef mixture into each bun.