Toss chicken, celery stalks, and 1 tsp. salt in a large bowl. Let sit until ready to serve.
Whisk buttermilk, vinegar, honey, 1 tsp. pepper, and remaining ½ tsp. salt in a small bowl or measuring cup. Pour over celery mixture. Add apricots, blue cheese, mint, and celery leaves and toss just to coat.
Transfer salad to a platter and season with more pepper.
