Place chicken in a large saucepan full of lightly salted water. Bring to a boil; boil for about 30 minutes, or until chicken is cooked through (juices run clear). Remove chicken from pan, reserving broth. Cut chicken into bite size pieces and place in bottom of a 9x13 inch baking dish.
Preheat oven to 350 degrees F (175 degrees C).
In a medium bowl mix together cream of chicken soup and cream of celery soup. Fill one empty soup can with milk, and mix milk with soups. Pour mixture over chicken. In a small bowl combine stuffing and broth; mix together and spoon mixture over casserole.