Preheat 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; stir in stuffing mix. Remove saucepan from heat, cover, and let sit until water is absorbed, about 5 minutes. Fluff stuffing with a fork.
Heat 1 tablespoon butter in a skillet over medium heat; cook and stir onion and celery until softened, 5 to 10 minutes.
Mix cream of chicken soup, sour cream, and cream of mushroom soup together in a bowl.
Spread shredded chicken into the bottom of the prepared baking dish. Layer onion mixture over chicken and season with salt and pepper; top with soup mixture. Spread stuffing over soup mixture layer.
Bake in the preheated oven until lightly browned and bubbling, about 1 hour.
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