Place chicken breasts in a large pot and add enough water to completely cover the chicken in liquid; bring to a gentle simmer. Cover tightly with a lid and cook until chicken is no longer pink in the center, about 15 minutes.
Remove chicken from the pot and cut into bite-sized pieces when cool enough to handle.
Mix together condensed soup and mayonnaise in a bowl.
Layer chicken in a 9x13-inch baking dish. Layer cooked broccoli over top, then soup mixture, and finally Cheddar cheese.
Sprinkle dry stuffing mix over top.
Bake in the preheated oven until bubbly and cheese has melted, 25 to 30 minutes.