Gather the flour into mound, make a well in the center, add the eggs to the well and work into a smooth dough. Allow the dough to rest for 20 minutes covered with a cotton kitchen towel or plastic wrap.
Roll the pasta into thin sheets 1/16” inch thick or run it through a pasta machine. Use the tip of a knife to cut the pasta into about thirty 3” x 4” rectangles. Bring a large pot of lightly salted water to the boil and quickly cook the pasta squares, 2 or 3 at a time, for about 30 seconds each.
As they cook, drain the pasta squares and line them up to cool on a wet kitchen towel that has been thoroughly wrung out. Pass the ricotta through a sieve, put in a bowl and add the drained and diced mozzarella. Cut the prosciutto into thin strips and add to the cheese mixture.
Season with salt and pepper, add the 2 eggs and work with a fork until the mixture is smooth. Melt half the butter in a pan over low heat, add the chopped tomatoes, a dash of salt and pepper, the basil and cook for 15 minutes.
Spread a tablespoon of filling on each of the squares of cooked pasta and roll them up to form cannelloni. Place the cannelloni in a single layer in a baking dish that has been buttered or lined with parchment paper.
Cover with the tomato sauce, grated Parmigiano-Reggiano and dot with the remaining butter. Bake for about 20 minutes in a preheated 350° oven and serve hot.