To Prepare Pasta: Make like basic pasta dough with flour, oil, vino santo and eggs in a well on a work surface. Roll on second thinnest setting of pasta machine and cut into lengths to match your lasagna pan. Boil in salted water for 1 minute and refresh in ice bath. Lay out on towels to dry.
To Prepare Filling: In a 12 inch to 14 inch saute pan, heat olive oil over medium heat until smoking. Add onion and pancetta and cook until soft and golden, about 8 to 10 minutes. Add chicken livers, gizzards and mushrooms and cook 8 to 10 minutes until light gray. Add tomato sauce, chicken stock and wine and cook over low heat for 1 hour until quite thick. Allow to cool.
To Prepare Lasagna: Preheat oven to 375 degrees F.
Brush a 14 inch by 9 inch lasagna pan and layer in the following order from the bottom: chicken liver sauce, pasta, bechamel sauce, with nutmeg added, and grated cheese (saving at least 1 cup bechamel sauce for last topping). Place Vincigrassi in oven and bake 35 to 40 minutes until bubbly and hot. Let stand 10 minutes and serve with grated cheese on top.
Saute the onion and garlic in the olive oil over medium heat until translucent, but not brown (about 10 minutes). Add the thyme and carrot and cook 5 minutes more. Add the tomatoes. Bring to a boil, lower the heat to just bubbling, stirring occasionally for 30 minutes. Season with salt to taste. Serve immediately, or set aside for further use.
The sauce may be refrigerated for up to one week or frozen for up to 6 months.