Cut the meats into pieces. Wash and roughly chop the vegetables, then mix them with the herbs and add to the meat, combining well. Place the meat and vegetables with the herbs in a large pot. Cook over medium heat with a lid, stirring occasionally. When the meat releases its juices, add salt, stir, and continue cooking until the mixture starts to brown but does not burn (about 20 minutes). Remove from heat and let cool slightly.

Meanwhile, make the béchamel sauce. In a saucepan, heat 6 1/3 cups milk (1.5 liters) until warm, then add 5 tablespoons all-purpose flour, whisking quickly to avoid lumps. Add salt and cook, stirring constantly, for 10 minutes after it starts to boil. At the end, add about 1 teaspoon ground nutmeg and 1 tablespoon butter. Set aside.

If making homemade pasta, after cutting it into rectangles (about 3 x 5 inches / 8 x 12 cm), cook the pieces for 2 minutes before assembling the cannelloni, then lay them out on clean kitchen towels. If using store-bought pasta sheets, cut them in half and soak two or three at a time in warm water before assembling. This helps prevent the pasta from absorbing too much béchamel and drying out, and ensures even cooking.

Once the meat mixture has cooled, blend it in a food processor, then transfer to a bowl and mix in about a quarter of the béchamel sauce until well combined.

On a cutting board, place half a pasta sheet (if using store-bought), spread a spoonful of the meat filling, and roll it up.

Place each roll side by side in a baking dish that has been greased with a mixture of butter, milk, and a little béchamel to prevent sticking.

After arranging all the cannelloni in the dish, pour the remaining béchamel sauce over the top, making sure to cover everything, and sprinkle generously with grated Parmesan cheese to cover all the cannelloni.