In a food processor, grind the carrot, onion, and fennel until fine.
In a large saucepan, heat the olive oil over low heat. Add the ground vegetables, pepper flakes, and garlic and simmer for about 30 minutes, or until the moisture evaporates, stirring occasionally.
In the same saucepan, brown the chicken and veal, then add the tomato sauce, wine, and veal stock. Stir to mix and simmer over medium to high heat, stirring occasionally, for 1 and 1/2 hours or until the sauce thickens and is reduced by one quarter. Remove from heat, stir in the butter and parmesan cheese and season with salt and pepper if needed; set aside.
To make the béchamel sauce: Combine 2 cups of the milk, the shallot, and nutmeg in a small saucepan and bring to a boil over medium heat. Reduce the heat and simmer, uncovered, for about 5 minutes. Cover and set aside to keep hot.
In another saucepan, melt the butter over medium to low heat. Add the flour and, using a wooden spoon, stir until the flour absorbs the butter and forms a roux. Continue stirring for 2 to 3 minutes longer while the mixture bubbles and thickens.
Slowly pour the hot milk into the flour, whisking continuously to prevent lumps. Bring the sauce to a boil over medium heat. Reduce the heat and simmer for 2 to 3 minutes, until smooth and thickened. Season to taste with salt.
Strain through a fine sieve into a bowl or glass measuring cup. Add as much of the remaining 1/2 cup milk as necessary to make 2 cups, whisking well to incorporate. Set aside, covered, until ready to use.
To make the sauce: In a large saucepan over medium heat, heat the olive oil. Add the garlic and onions, and cook, stirring, for 3 minutes, or until the garlic just begins to turn golden brown.
Increase the heat to medium-high, add the tomatoes, red pepper flakes, and salt and pepper, and cook, stirring occasionally, for 15 minutes, or until the tomatoes are softened and a sauce is formed. Stir in the fresh basil and remove from heat.
To make the lasagna: Bring a large pot of lightly salted water to a boil over high heat and cook the pasta until al dente, about 1 minute. Drain and shock in cold water.
Preheat oven 350°F. In the bottom of 8 x 12-inch baking pan, spray with non stick baking condiment of your choice or grease with butter. Cover the bottom of the baking dish with one layer of pasta sheets.
Using a spatula, spread 1 cup prepared béchamel sauce evenly across bottom, next on top of béchamel sauce spread evenly approximately 1 cup prepared Bolognese sauce, then sprinkle these two layers with approximately ½ cup parmesan cheese. Place second pasta sheet over filling and press down evenly.
Repeat two more times to yield 3 layers and cover the top of the lasagna with pasta sheets.
Spread remaining béchamel and ½ cup tomato sauce on top layer of lasagna and sprinkle with 1/2 cup grated parmesan cheese.
Bake lasagna for 1 hour until bubbling or until the top is golden. Remove from oven and let set stand for a few minutes to set before cutting. Serve with remaining parmesan cheese.