Heat butter and oil in a saucepan until butter melts. Add flour and cook for two minutes, stirring constantly. Gradually pour in milk, broth, salt, pepper and nutmeg and cook over medium heat, stirring constantly, until thickened and smooth. Add chopped eggs and parmesan cheese and cook over low heat about 5 minutes more. Check seasoning. Spread sauce on a dinner-size plate, refrigerate until firm - at least one hour. Recipe may be prepared ahead up to this point.
Coat croquettes with beaten egg and roll in breadcrumbs. Refrigerate if you're not going to cook them right away. Fry quickly in hot oil (about 380 degrees F) at least 1/2-inch deep - until golden. Or, better yet, use a deep fryer. Croquettes may be kept warm in 200 degrees F oven up to 30 minutes.