For the pasta, on a large work surface, combine the flour and salt; make a well. Add the whole eggs, egg yolk, water and EVOO to the well and mix with a fork. Using the fork, start pulling the flour into the well to make a sticky dough, then work in the flour a little at a time with your hands to fully incorporate. Knead the dough in one direction away from your body, turning it into a smooth and elastic dough, 5-10 minutes. Cover with a kitchen towel and let stand for about 30 minutes.
Cut the dough into 4 equal pieces with a bench scraper. Working with one piece of dough at a time (keep the dough that you are not working with under a clean kitchen towel to keep it from drying out), set your pasta machine to the widest setting and pass the dough through. Fold the dough over on itself in thirds and pass through the pasta machine three more times. Next, pass the dough through the machine decreasing the width with each pass until the dough is at the second to last setting, forming a 1/16-inch thick sheet. Cut the long sheet in half crosswise and place the sheets on parchment paper dusted with flour. Repeat this process with each piece of dough.
For the lasagna, use the whole sheet or cut each sheet crosswise into thirds if it is easier to layer them this way.
Make the ragu: In a large skillet or Dutch oven, heat the oil, 2 turns of the pan, over medium heat. When it ripples, melt in the butter. Add the carrot, celery, onion and garlic; partially cover and cook until the vegetables are softened, 7-8 minutes. Add the meat, sage, rosemary, bay leaf and cloves, and season with salt and pepper. Cook, breaking up the meat into crumbles as it browns. Deglaze the pan with the wine, let it cook off then add the tomato paste and cook until it has been absorbed. Add stock and simmer for 30 minutes. Discard the bay leaf.
Preheat the oven to 375°F.
Make the besciamella: In a saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1 minute. Whisk in the milk and season with salt, pepper and a little nutmeg. Cook the sauce until thick enough to coat the back of a spoon.
If using fresh pasta sheets, heat a large and wide pot of water to a low rolling boil. Fill a pan with ice and water. Salt the boiling water and cook the pasta sheets 2 at a time for about 2 minutes. Cold-shock the sheets in the ice water to stop the cooking process. Drain on a clean kitchen towel and transfer to parchment paper. Stack the sheets neatly (you can do this ahead and freeze or store in the fridge until ready to use).
Coat the bottom of a 9x13” lasagna pan or two 8” square pans with a thin layer of besciamella. Add a pasta sheet or sheets, top with the besciamella and the ragu in thin layers, and sprinkle with cheese. Repeat the layering to the top of the pan(s), ending with besciamella and cheese. Bake until browned and bubbling, 30-35 minutes (1 hour if using dried pasta). Top with parsley and let stand for 20 minutes before cutting and serving.