Fill a large pot with lightly salted water and bring to a rapid boil. Add egg noodles and cook at a boil until tender yet firm to the bite, 7 to 9 minutes. Drain.

Combine drained noodles, chicken, ham, and Swiss cheese in a large mixing bowl; mix until well combined.

Stir together cream of chicken soup, milk, and sour cream in a separate bowl until smooth. Pour sauce over noodles and mix until sauce is well distributed. Transfer mixture to a large microwave-safe casserole dish.

Place butter in a microwave-safe bowl. Cook on high until melted, about 30 to 45 seconds. Add bread crumbs and Parmesan cheese. Mix until well blended and crumbly and sprinkle on top of casserole.