Heat the water and chicken stock in a medium saucepan to boiling. Add the rice and reduce the heat. Simmer, partially covered, until most of the liquid is absorbed, 15 to 20 minutes. Stir in the egg, butter, and cheese. Spoon the rice into a bowl or on a baking sheet and refrigerate.
For the filling, soak the porcini in the hot water for an hour. Drain, reserving the liquid, and finely chop the mushrooms. Strain the soaking liquid.
Heat the olive oil in a sauté pan or skillet over medium heat. Add the onion and cook, stirring occasionally, until translucent, about 10 minutes. Stir in the garlic and cook a few minutes. Add the porcini and the soaking liquid, the prosciutto, and tomato puree; simmer until thickened, 15 to 20 minutes. Season to taste, then cool.
To assemble the suppli, take a few tablespoons rice in your hand and make a pocket in the center with your finger, cupping the rice in your hand. The rice layer should be about
About 15 minutes before serving, heat peanut oil in a deep fryer to about 350°F. Add several croquettes and fry 3 minutes. Remove and let rest a minute or two, then return to the hot oil and fry until golden, about 3 more minutes. (This allows the mozzarella to melt.) Drain on paper towels and repeat with the remaining croquettes. Serve hot.