1. Make the Meat Sauce: Heat olive oil in a large saucepan over medium-high heat. Add carrot, celery, and onion, and cook for 2 minutes. Add beef and pork, and cook, stirring occasionally, until meat is nicely browned, 15 to 20 minutes. Add tomatoes, tomato paste, and 1/4 cup water. Season with salt and pepper. Cook, covered, over low heat, stirring occasionally, for 1 hour. Remove from heat, and set aside.

2. Make the Lasagna Dough: In a food processor, combine flour, eggs, oil, and salt. Process until it forms a soft, but not sticky dough. If dough is sticky, it may be necessary to add a little more flour. Wrap dough with plastic wrap, and let rest for 30 minutes.

3. 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.

4. To assemble: Bring a large pot of salted water to a boil. Using a pasta machine, roll dough into long sheets as thin as possible. Let sheets dry for a few minutes. Then cut them into sections, each about 13 inches long. Add oil and lasagna sheets to boiling water. Cook, stirring gently to prevent them from sticking to one another, for about 2 minutes. Drain, and rinse under cold running water. Lay sheets out on a clean kitchen towel, and set aside until ready to use. Be sure not to let them stand for more than 30 minutes.

5. 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.