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Trim fat from meat. If necessary, cut meat to fit into a 3 ½- or 4-quart slow cooker. Place onion in bottom of slow cooker. Add meat; sprinkle with thyme and rosemary. Pour broth over meat.
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Cover and cook on low-heat setting for 9 to 10 hours or on high-heat setting for 4 ½ to 5 hours.
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Meanwhile, for barbecue sauce, in a medium saucepan, combine vinegar, ketchup, brown sugar, honey, Worcestershire sauce, mustard, garlic, black pepper, and salt. Bring to boiling; reduce heat. Simmer, uncovered, for 20 to 25 minutes or until slightly thickened, stirring occasionally.
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Transfer meat to a cutting board. Using two forks, pull meat apart into shreds, discarding fat. Strain onions, discarding liquid. Return shredded meat and onions to cooker. Stir in barbecue sauce. If using high-heat setting, turn to low-heat setting Cover and cook for 1 hour.
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To serve, spoon meat mixture onto bun bottoms. Top with coleslaw. Add bun tops.