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, six to eight minutes per side. Transfer to a cutting board and slice into 1/2 – inch thick strips. Add to hot-sauce mixture and turn to coat. Drizzle dressing over salad and toss until coated. Top with chicken and cheese.
I didn’t happen to have romaine, so I used iceberg. Also, no blue cheese, so feta was substituted. We like things hot, so I didn’t put the butter or honey in the hot sauce. I used Frank’s Red Hot, right out of the bottle.