Boil chicken breasts in a pot of lightly salted water until no longer pink in the center and are beginning to fall apart, about 30 minutes. Remove chicken breasts from water using a slotted spoon and shred.
Preheat oven to 350 degrees F (175 degrees C).
Melt 1 tablespoon butter in a skillet over medium heat; cook and stir onion in the melted butter until translucent, about 10 minutes. Add shredded chicken; mix well. Pour cream of celery soup and evaporated milk into chicken mixture; spoon mixture into an 8x8-inch baking dish.
Mix chicken broth and melted butter together in a bowl. Empty cornbread stuffing mix into a large bowl; pour chicken broth mixture over stuffing. Toss to combine. Pour stuffing over chicken mixture.
Bake in the preheated oven until bubbling and stuffing is lightly browned, about 45 minutes.
