Preheat oven to 350 degrees F (175 degrees C). Lightly grease a 9×13-inch casserole dish.
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 and transfer noodles to the prepared casserole dish; top with chicken, ham, and Swiss cheese.
Mix cream of chicken soup, milk, and sour cream together in a bowl; spoon soup mixture over noodle mixture.
Melt butter in a skillet over medium heat; cook and stir bread crumbs and Parmesan cheese in the melted butter until crumbs are coated with butter and cheese, 2 to 3 minutes. Sprinkle bread crumb mixture over casserole.
Bake in the preheated oven until casserole is bubbling and lightly browned, about 30 minutes. Allow casserole to cool for 10 minutes before serving.
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