Preheat the oven to 375°F (190°C). Spray a 9x13-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 from heat, cover, and let sit until water is absorbed (about 5 minutes). Fluff stuffing with a fork.

Melt 1 tablespoon butter in a skillet over medium heat. Add onion and celery; cook and stir until softened (5–10 minutes).

Mix cream of chicken soup, cream of mushroom soup, and sour cream in a bowl until combined.

Spread shredded chicken at the bottom of the prepared baking dish. Layer the onion-celery mixture over the chicken and season with salt and pepper.

Top with the soup-sour cream mixture and evenly spread stuffing over the top.

Bake uncovered in the oven until bubbly and golden-brown (about 1 hour).