Trim the chicken and slice each breast horizontally to make thinner cutlets for even cooking. Lightly season with a small pinch of salt, then refrigerate while you mix the sauce (about 10 minutes).
In a large measuring cup or bowl, whisk together the barbecue sauce, honey, apple cider vinegar, Worcestershire, smoked paprika, and minced garlic until smooth and glossy.
Spread about 1/3 cup of the sauce in the bottom of the slow cooker to prevent sticking. Arrange the chicken pieces in a snug single layer and pour the remaining sauce over the top, coating well.
Cover and cook on High for 3 to 3 1/2 hours or on Low for 5 to 6 hours, until the chicken registers 165°F and is very tender. If your slow cooker runs hot, crack the lid slightly during the last 30 minutes.
Transfer the chicken to a cutting board. Switch the slow cooker to High and simmer the sauce with the lid off for 10 to 15 minutes to thicken, stirring once or twice. Shred the chicken with two forks while the sauce reduces.
Return the shredded chicken to the cooker and toss until every strand is coated and glossy. For caramelized edges, spread the chicken on a foil-lined sheet pan and broil for 3 to 4 minutes, then return to the cooker to keep warm.
Serve on toasted buns, over rice, or piled onto baked potatoes. Tip: If the sauce tastes too sweet, stir in 1 teaspoon vinegar; if too tangy, add 1 to 2 teaspoons honey.