Preheat a grill for indirect cooking over medium-high-heat (350 to 400 degrees F).
Combine the buttermilk, blue cheese, vinegar and Worcestershire in a medium bowl; add the carrots and celery and toss thoroughly. Allow to sit at room temperature while you cook the chicken.
Combine the hot sauce and butter in a small saucepan and bring to a boil; cook until slightly thickened, 1 to 2 minutes.
Meanwhile, grill the chicken, meaty-side up, and cook until the underside is well-marked and readily releases from the surface, about 15 minutes. Turn and cook until a thermometer inserted in the thickest part of the meat (not touching bone) registers 165 degrees F, about 15 more minutes. Brush liberally with the sauce, then turn the chicken and brush the tops with more sauce. Grill 1 more minute. Serve with the slaw.