In a mixing bowl, stir together the beef, milk, parsley, salt, and pepper. Form into balls the size of olives. Heat 2 tablespoons of olive oil in a skillet and brown the meatballs in small batches. Remove from the pan as they brown and drain on paper towels. Set aside.
In the same skillet, add the onion and garlic and saute until the onion is lightly browned. Then stir in the tomato puree and tomato paste. Simmer for 15 minutes.
Bring 6 quarts of water to boil in a large pot.
Add the meatballs to the tomato mixture and continue cooking for another 30 minutes. Meanwhile, liberally salt the boiling water and add the lasagna. Cook until al dente, about 10 minutes. Drain in colander.
Preheat the oven to 375 degrees F.
In a shallow ovenproof pan, roughly 13 by 9 by 2 inches, spread a thin layer of the sauce (no meatballs). Then spread a layer of overlapping lasagna 1 strip thick (don't let the strips run up the side of the dish). Cover that with mozzarella slices and then 5 tablespoons ricotta. Sprinkle with the Parmesan and then spread on 1/4 of the sauce and meatballs. Begin again with a layer of lasagna and continue as above until all the ingredients are used up, ending with the Parmesan.
Bake for 30 to 35 minutes. If the cheese on top hasn't melted, run under the broiler briefly. Then let the dish rest at room temperature for a few minutes before serving.