Mix mayonnaise, 1 tablespoon of the Seasoning, vinegar, brown sugar and milk in medium bowl until well blended. Refrigerate until ready to serve.
Brush chicken with oil. Sprinkle both sides evenly with remaining 1/4 cup of the Seasoning. Grill chicken over medium heat 6 to 8 minutes per side or until cooked through. Remove chicken from grill. Transfer to large cutting board or bowl.
Shred chicken using 2 forks. Serve pulled chicken on rolls, topped with Alabama-Style BBQ Sauce and pickles.