Mix chicken broth, tomato paste, garlic, brown sugar, Worcestershire sauce, garlic salt, vinegar, cumin, paprika, and cayenne pepper together for sauce in a small bowl.
Place chicken in a 3-quart slow cooker. Pour sauce over top.
Cover and cook until chicken is no longer pink in the center and the juices run clear, on Low for 5 hours or on High for 4 to 5 hours. Remove chicken and turn the slow cooker to Warm.
Shred the chicken with 2 forks and return to the slow cooker. Heat for 10 minutes to allow juices to soak into the shredded chicken.