Preheat oven to 375 degrees F. Boil chicken in a large pot for 25 minutes. Prepare stuffing according to box directions on stove. Steam broccoli 10 to 15 minutes. Dump both cans of soup into a bowl. Whisk soup until well blended. Pour milk into cans, approx 3/4 of each can. Take whisk or knife and stir milk in can. Pour milk into soup bowl. Whisk again. Do not cook soup.
Now, strain chicken into a pasta strainer. Cut up chicken into small to medium chunks. Spread chicken into a 10x15x2-inch cake pan. Sprinkle some of the cheese over the top of the chicken. Spread broccoli pieces over chicken, and then pour most of the soup over broccoli. (Keep about a 1/2 cup for last step). Now sprinkle the remaining cheese over the broccoli. Place prepared stuffing over the entire top. Do not press down. Pour the remaining soup over top of stuffing. Do not cover the stuffing entirely with soup. You want to just pour slight strips of soup.
Put into oven and bake 15 minutes. Let set 10 minutes.