Trim fat from meat. If necessary, cut meat to fit into a 3 1/2 - 4 quart slow cooker. In the cooker, combine onion and green pepper. Add meat; sprinkle with thyme and rosemary. Pour broth over meat. Cover and cook on low for 7-8 hours (time will vary by slow cooker).
Meanwhile, make the barbeque sauce. In a medium saucepan, combine vinegar, ketchup, brown sugar, honey, Worcestershire sauce, mustard, chili pepper, garlic, salt, and pepper. Bring to a boil, reduce heat. Let simmer, uncovered, for 20-25 minutes, or until thickened, stirring occasionally.
Transfer meat to a cutting board. Using two forks, pull meat apart into shreds, discarding fat. Strain vegetable mixture, discarding liquid. Return meat and strained vegetables to the slow cooker. Stir in barbeque sauce. Cover and cook on low for 30 minutes.