Soak the mushrooms in 1/2 cup water for 2 hours. Drain the mushrooms, reserving the mushroom water, and chop finely. Dissolve the tomato paste in the mushroom water and set aside.
Meanwhile, in a medium saucepan, combine the beef stock, crushed tomatoes and 3 tablespoons butter, mix well and bring to a boil. Add the rice, reduce the heat to a simmer and cook until the rice is done, stirring occasionally. Turn the rice mixture out into a serving bowl and gradually stir in the eggs and Parmigiano. Set aside to cool.
Meanwhile, in a 12 to 14-inch saute pan, melt the remaining tablespoon of butter over low heat. Add the onion and prosciutto and cook over high heat 3 minutes. Add the ground beef and cook until well browned. Add the mushrooms, mushroom water-tomato paste mixture, and salt and pepper to taste, and keep at a simmer. With a spoon, make egg-shaped portions of the rice, and make an indentation in the center of each one. Place a teaspoon of filling in the center of each ball of rice, along with a few cubes of the cheese. Roll each ball in the bread crumbs so that it is completely coated, and set aside.
In a large, heavy bottomed pot, heat the olive oil over high heat until it is almost smoking 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.
Recipe copyright 2000, Mario Batali. All Rights Reserved.