To make the meat sauce, heat the butter in a skillet over moderate heat and saute the onion until tender but not brown, about 5 minutes.
Add the beef and lamb and saute until browned.
Add the remaining ingredients and bring to a simmer.
Cook covered over low heat, stirring occasionally, for 30 minutes.
Meanwhile, to make the béchamel sauce, heat the butter in a saucepan over moderate heat.
Stir in the flour and cook for 3 minutes.
Stir in the milk and bring to a boil, stirring frequently.
Remove from the heat and stir some of the milk mixture into the beaten eggs.
Stir the egg mixture into the milk mixture and season with nutmeg, salt, and pepper.
Cook the pasta in salted water until slightly under-done, about 8 minutes.
Drain and place half the cooked pasta in a greased 3-quart (3 L) baking dish.
Sprinkle with half the cheese and spread all of the meat sauce over the top.
Add the remaining pasta, all of the béchamel sauce, and the remaining cheese.
Bake in a preheated 350F (180C) oven until the top is browned and bubbling, 45 minutes to 1 hour.
Let stand at room temperature for 15 minutes before cutting into serving portions.