Lasagna with Ragu Bolognese

3 tablespoons unsalted butter3 tablespoons extra-virgin olive oil2 onions, cut into 1/4-inch dice3 stalks celery, cut into 1/4-inch dice3 carrots, cut into 1/4-inch dice2 pounds ground sirloin2 pounds ground veal1 quart whole milk1 quart Homemade Beef Stock, or two 14 1/2 -ounce cans1 cup tomato paste1 1/2 teaspoons coarse salt3/4 teaspoon freshly ground pepper In a large cast-iron or enamel pot, heat butter and olive oil over medium heat. Add onions, and cook until they begin to soften, about 5 minutes. Add celery and carrots, and cook until the vegetables are tender, 8 to 10 minutes. Add ground sirloin and veal, and cook, stirring occasionally, until the meat is no longer pink. Add milk, and cook at a gentle simmer, skimming fat from surface, until the liquid has reduced by half, about 50 minutes. Add beef stock, tomato paste, salt, and pepper; simmer gently until sauce thickens, 40 to 45 minutes. If using for lasagna, set aside to cool slightly before assembling. The sauce can be stored in an airtight container in the refrigerator for up to 3 days, or in the freezer for 2 months. 4 tablespoons unsalted butter, cut into small pieces, plus more for baking dish3 tablespoons all-purpose flourPinch freshly grated nutmeg1 cup milk1 tablespoon olive oil3/4 cup grated Parmesan cheese1 tablespoon unsalted butter, cut into small pieces, plus more for baking dish Make the Bechamel Sauce: In a small saucepan, combine butter and flour. Stir in nutmeg, and season with salt and pepper. Place over medium heat and stir in milk. Bring to a boil. Reduce heat to medium-low, and continue cooking until the mixture begins to thicken, about 2 minutes. Remove from heat, and set aside. Preheat the oven to 350 degrees. Lightly butter a 13-by-9-inch baking dish. Spoon a little of the meat sauce into the bottom of the prepared baking dish, and spread to coat. Then, layer with lasagna sheets, one-sixth of the bechamel sauce, one-sixth of the meat sauce, and a little Parmesan. Repeat the layers five more times. Top with remaining Parmesan and pieces of butter. Bake until cheese begins to turn golden brown, 30 to 45 minutes. Serve warm.