Bring a large pot of lightly salted water to a boil. Cook egg noodles in the boiling water, stirring occasionally until cooked through but firm to the bite, about 5 minutes. Drain.
Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Mix cream of chicken soup and sour cream together in a bowl; stir in chicken and noodles. Spoon mixture into the prepared baking dish; sprinkle with stuffing mix.
Combine chicken broth and butter in a saucepan over medium heat until butter is melted; pour over stuffing layer.
Bake in the preheated oven until casserole is bubbling and stuffing is browned, about 30 minutes.