Add the onions, carrots, celery, and mushrooms and cook until soft, about 5 minutes. Add the garlic, cloves and nutmeg to the pan and cook for 2 minutes. Add tomato paste and cook for 2 minutes. Add the wine and cook until almost evaporated.
Preheat the oven to 350 degrees.
Butter a large rectangular baking dish, then spoon 1/2-cup of meat sauce onto the bottom of the dish. Cover with 1 sheet of fresh pasta. Top the lasagna with a layer of meat sauce (making certain that pasta is completely covered), a layer of bechamel, then a light dusting of cheese. Repeat layering lasagna, sauces, and cheese in this manner until all have been used, ending with a topping of bechamel and cheese.
For Bechamel: In a saucepan, melt the butter over low heat and stir in the flour, stirring constantly until smooth, about 2 minutes. Slowly whisk the milk into the flour, stirring vigorously to blend together. Set over high heat and quickly bring to a boil for 1 minute, stirring. Allow to cook another 5 minutes, or until floury taste is gone. Remove from heat and add salt and nutmeg to taste.