Make the bolognese sauce:
1. In a medium dutch oven, or other heavy-bottomed pot, melt one tbsp of the butter with the olive oil over medium heat. Add the onion, celery, carrot and garlic and cook, stirring frequently until vegetables are soft, and onions are translucent, 6-8 minutes.
2. Add the beef, pork and veal and raise the heat to medium high. Cook, stirring frequently, until meat browns slightly, 8-10 minutes. Add the tomato paste and stir to combine. Cook for two more minutes to combine the flavors.
3. Add the wine, 6 oz (3/4 cup) of the milk and sprig of thyme and stir to combine, bring the sauce to a boil, and reduce heat to low to maintain a gentle simmer. Cover and cook for 1 1/2 to 2 hours, stirring occasionally, until the mixture comes together as a sauce. This can be made ahead and refrigerated or frozen. Warm slightly before assembling the lasagna.
Make the Pasta:
4. Pile 2 1/4 cups of the flour in a large bowl, and make a well in the center so the flour looks like a crater. Crack the eggs in the middle of the well.
5. Using a fork, beat the eggs lightly to crack the yolks, and then slowly begin pulling the flour into the eggs from the walls of the well, without collapsing the walls entirely. Use the fork to mix the flour into the eggs until a slightly stiff, shaggy dough forms.
6. When there is no longer any liquid to incorporate, and the fork is becoming difficult to use, use your hands to begin to knead the dough into a ball. You want to continue mixing in the flour, but not so much that the dough is unworkable and stiff. Once you've incorporated as much flour as you can, remove the dough from the bowl to a very lightly floured surface. You will likely have 1/4 to 1/2 cup flour left in the bowl. Wipe your hands clean of any flour and bits of dough.
7. With clean, lightly floured hands, knead the dough by pressing away from you with the heel of your hand and folding the dough back over on itself towards you, without adding any additional flour, if possible, for 8 - 10 minutes until the dough is very smooth and elastic. Cover the dough loosely, and let rest for about 30 minutes.
8. Cut the dough into four pieces and roll each piece through the pasta machine, until as thin and translucent as possible - usually to the second thinnest setting on the machine. I find it works best if you roll the dough through the thickest setting five or six times, until the dough feels like leather, then through each of the thinner settings twice before moving to the next one. When you have finished rolling each piece of dough, cover with a kitchen towel as you work with the next piece. Set aside until you are ready to assemble the lasagna.
Make the Bechamel:
9. Heat 3 cups (24 oz) of the milk to a simmer in a small saucepan over low heat, or in the microwave.
10. Melt 6 tbsp of the butter in a saucepan over medium low heat. Add 6 tbsp of the flour and stir until the mixture thickens and starts to turn golden.
11. Whisk in the simmering milk, raise the heat to medium and cook, stirring constantly, until the mixture is thick and bubbly, 6-8 minutes. Season with salt and pepper and turn off the heat.
Assemble the Lasagna:
12. Preheat the oven to 350 degrees.
13. Lightly cover the bottom of a 9" square baking pan with a thin layer of the bolognese sauce. Lay sheets of pasta, cut to fit the pan and overlapping slightly if necessary, to cover the bottom of the pan. Layer approximately a 1/2 cup of the bolognese over the pasta sheets and use a spoon to spread the sauce to cover the pasta. Follow with a 1/2 cup of the bechamel, and spread with a spoon.
14. Repeat the pasta, bolognese and bechamel four more times, until you have five complete pasta/bolognese/bechamel layers. Cover with a final layer of pasta, and another 1/2 cup of the bechamel. Sprinkle the parmesan cheese over the top.
15. Cook the lasagna for 30 minutes at 350 degrees, then turn the heat up to 400 and cook for another 15 minutes, until the lasagna is browned and bubbly. Remove from the oven and let sit for 10 minutes before slicing and serving. Enjoy!