Preheat barbecue to medium. Peel long ribbons from carrot into a large bowl, turning as you go. Add lettuce leaves and celery. Whisk milk with mayo, dill, vinegar and garlic in a small bowl. Stir hot sauce with butter and honey in a large bowl. Set both aside.
Oil grill, then barbecue chicken, covered, until springy when pressed, 6 to 8 min per side. Transfer to a cutting board and slice into 1/2-in. thick strips. Add to hot-sauce mixture and turn to coat. Drizzle dressing over salad and toss until coated. Top with chicken and cheese.