Instead of fresh mushrooms you can use 1 3/4 ounces of dried mushrooms. If using dried mushrooms, soak them in lukewarm water for 20 minutes before beginning to prepare the rest of the dish. Gather the flour into a mound, make a well in the center, add the eggs and work into a smooth dough.
Let rest for 20 minutes covered by a kitchen towel or plastic wrap. Roll the pasta into thin sheets 1/16” thick with a rolling pin or a pasta machine. With a sharp knife, cut the pasta into about thirty squares 3/8” on a side. In a pot, bring a large quantity of lightly salted water to a boil and drop in the pasta squares 2 or 3 at a time, cooking each one for about 30 seconds.
Remove and drain the pasta squares, allowing them to cool on a damp kitchen towel that has been wetted and thoroughly wrung out. Separately, prepare a béchamel sauce. In a small skillet melt the butter, add the sifted flour and cook over low heat for 1 minute, stirring with a whisk. Gradually add the milk and bring to a boil, stirring constantly. Salt to taste and season with a little freshly grated nutmeg.
Continue cooking for a few minutes until thickened. In the meantime, if you are using fresh mushrooms, carefully clean,trim off the root end of the stem and scrub the surface with a wet brush, then cut into thin, even slices. Put the butter in a saucepan over medium heat. When the butter melts, add the fresh or dried mushrooms with 3 or 4 tablespoons of lukewarm water and the salt and cook over high heat until the water has evaporated.
When the mushrooms are done, put about 2/3 of the béchamel, which has cooled in the meantime, into a bowl and add the cup of Parmigiano and mushrooms. Use a tablespoon to spread this filling over each of the cooked pasta squares and roll up the pasta into cannelloni.
I Place the cannelloni in a single layer in an ovenproof baking dish that has been thoroughly buttered or lined with parchment paper. Dilute the reserved béchamel by adding one cup of milk and heat it on the stove for a few minutes, stirring often.
Pour this sauce over the cannelloni, cut the Gruyère and prosciutto into small cubes and sprinkle over the cannelloni. Sprinkle the entire dish with the grated Parmigiano. Cook for about 20 minutes in a preheated 350° oven and cook until the sauce covering the cannelloni is lightly browned.