Soak the mushrooms in hot water for 30 minutes. Drain the mushrooms, reserving the mushroom water, and chop finely. Strain soaking liquid thrugh a fine sieve into a small bowl and set aside.
Meanwhile, in a medium saucepan, heat 3 tablespoons olive oil and butter over medium heat. Add the onion and prosciutto and cook until onion is softened, about 7 minutes. Add the rice and cook, stirring until the rice is opaque, about 3 minutes. Add 2 cups hot water and mushroom water, bring to a boil, and cook uncovered, until liquid is absorbed, about 20 minutes. Turn the rice mixture out into a serving bowl and allow to cool 10 minutes. Stir in the eggs, parsley, and Parmigiano.
For filling, mix the chopped porcini and mozzarella cubes together in a bowl.
Using a tablespoon, make egg-shaped balls of the rice mixture, about 12-14. Use thumb to make an insentation in the center of each, insert 1 teaspoon filling, use palms to round rice around filling so it is completely enclosed.
In a large, heavy bottomed pot, heat the 4 cups olive oil over high heat until it is almost smoking. Roll each ball in bread crumbs and set aside on a plate. Working in batches, fry the balls in the hot oil until they are golden brown, about 5 minutes. Remove with a slotted spoon and allow to drain on paper towels while seasoning with salt and pepper. Serve with tomato sauce, if desired.