Combine the rice and broth in a large pot. Bring to a boil and then reduce to a simmer, cover, and cook until the water is absorbed and the rice is soft and creamy, about 15 to 20 minutes. Stir occasionally to ensure the rice doesn't stick to the bottom of the pot.
While the rice is cooking, heat the butter and olive oil in a small pan over medium heat and add the prosciutto. Cook until it begins to crisp, about 2 minutes. Set aside.
Once the rice is cooked, remove to a large baking sheet and let cool for 10 minutes. (Rice can be made a day ahead, cooled, and stored in the refrigerator overnight.)
While the rice is cooling, pour 1/2 cup (120 ml) room-temperature water into a shallow bowl. Place the bread crumbs in a separate shallow bowl. Set both aside.
Once the rice is cool, combine it with the crisped prosciutto, egg, parsley, Parmigiano cheese, and salt in a large bowl.
Using an ice cream scoop or lightly wet hands, create a 2-inch (5-cm) round ball with the rice mixture. Make an indentation in the middle and add a cube of mozzarella cheese. Encase the cheese with rice and reshape into a perfect ball, then set on the baking sheet. Repeat with the remaining rice mixture.
Using your hands, lightly wet the outside of each of the rice balls with the room-temperature water, then roll in the bread crumbs. Set them on the baking sheet.
Heat the vegetable oil in a large pot to 350 degrees F (175 degrees C). Drop 4 rice balls into the oil (they should fully submerge) and cook until the outside is golden brown, about 3 minutes. If your pot is wide, carefully roll the rice balls in the oil to ensure even cooking on all sides.
Transfer to paper towels to drain. Repeat with the remaining rice balls. They are best eaten at room temperature, garnished with a sprinkling of grated parmesan cheese, if you can wait!