Steam rice and butter together, sprinkling parsley. Put in bowl and set aside for cooling. Beat two of the eggs. Stir eggs into the cooled steamed rice until well mixed.
Use hands to form the rice mixture into balls the size of a large egg. If the mixture is too moist to hold its shape well, stir in a few spoonfuls of bread crumbs. Poke a hole in the center of each ball with your finger, then fill it with small cubes of mozzarella and close the hole over again with the rice mixture.
Heat the oil for deep-frying until a small piece of bread sizzles as soon as it is dropped in. Spread out some flour on a plate. Beat the remaining egg in a shallow bowl. Sprinkle another plate with breadcrumbs. Roll the rice balls in the flour, then in the egg, then in the breadcrumbs.
Deep-fry the covered rice balls, a few at a time, in the hot oil until golden and crisp. Drain on paper towel, and serve.