1. In a small bowl, soak the mushrooms in the hot water for 30 minutes.
2. Lift out the mushrooms, reserving the liquid, and finely chop. Strain the soaking liquid through a fine sieve into another small bowl and set aside.
3. In a medium saucepan, heat the 3 tablespoons olive oil and butter over medium heat. Add the onion and prosciutto and cook until the onion is softened, about 7 minutes. Add the rice and cook, stirring until opaque, about 3 minutes. Add 2 cups hot water and the mushroom water, bring to a boil, and cook uncovered, until the liquid is absorbed, about 20 minutes. Turn the rice out into a bowl, and allow to cool for 10 minutes. Stir the eggs, parsley, and Parmigiano into the rice.
4. For the filling, mix the chopped porcini and the mozzarella cubes together in a bowl.
5. Using a tablespoon, make egg-shaped balls of the rice mixture; you should have 12 to 14. Use your thumb to make an indentation in the center of each, insert 1 teaspoon of the filling, and use your palms to round the ball so that the filling is completely enclosed.
6. In a large deep pot, heat the 4 cups olive oil over medium-high heat until it reaches 370°F. Roll each ball in the bread crumbs so that it is completely coated, and set on a plate. Working in batches, fry the balls in the hot oil until golden brown, about 5 minutes. With a slotted spoon, transfer to paper towels to drain, and season with salt and pepper. Serve warm or at room temperature (do not refrigerate).