Cook noodles in a large pot as package directs. Transfer to a wide, shallow bowl.
Meanwhile, in a large wide pot over high heat, cook mushrooms, onion, and eggplant in 2 tbsp. each butter and oil, stirring often, until softened, about 5 minutes, adding a little more oil if needed.
Stir in salt, pepper, and flour and cook 1 minute. Add broth and milk, reduce heat, and simmer 5 minutes.
Put sour cream and nutmeg in a bowl. Stir in hot broth mixture 1/4 cup at a time, then return to pot. Spoon eggplant sauce over noodles and sprinkle with parsley.