Preheat the oven to 375 degrees F (190 degrees C). Spray a 9×13-inch baking dish with cooking spray.
Bring water and 1/4 cup butter to a boil in a saucepan over medium heat. Add stuffing mix and stir to combine. Remove the saucepan from the heat, cover, and let sit until water is absorbed, about 5 minutes. Fluff stuffing with a fork.
While you are preparing stuffing, melt 1 tablespoon butter in a skillet over medium heat. Add onion and celery; cook and stir until softened, 5 to 10 minutes.
Mix both condensed soups with sour cream in a bowl until well combined.
Spread shredded chicken in the bottom of the prepared baking dish. Layer onion mixture over chicken and season with salt and pepper.
Top with soup mixture and spread stuffing evenly over top.
Bake in the preheated oven until filling is bubbly and topping is golden brown, about 1 hour.