Preheat oven to 375°F (190°C).
Cook noodles: Bring a large pot of salted water to a boil. Cook the egg noodles until just al dente, about 7 minutes. Drain and set aside.
Mix filling: In a large bowl, whisk together the cream of chicken soup, milk, cheddar cheese, pepper, and salt.
Add ingredients: Stir in the shredded chicken, chopped onion, celery, and thawed peas and carrots. Gently fold in the cooked noodles until everything is evenly mixed.
Assemble: Transfer the mixture into a 9×13-inch baking dish, spreading it evenly.
Make topping: In a separate bowl, toss the crushed butter crackers with the melted butter. Sprinkle this mixture evenly over the casserole.
Bake: Place the dish in the oven and bake for 30 to 35 minutes, or until bubbly and golden brown on top.
Garnish & serve: Remove from the oven and garnish with chopped parsley before serving.