In a Dutch oven, bring 3 cups of water, the salt, and the butter to a boil. Add the rice and simmer uncovered, stirring frequently, until all the water is absorbed, about 20 minutes.
Stir the eggs, cheese, parsley, and garlic into the rice. Spread the rice onto a large baking sheet and refrigerate for 30 minutes.
In a large bowl mix together the flour with a cup of water, and whisk it until the lumps are gone. Fill a second bowl with the breadcrumbs.
Form the rice into balls about the size of a clementine. Pick up one ball at a time and rub it all over with some of the flour and water mixture, and then roll it in the breadcrumbs until it is completely coated.
In a Dutch oven, or large pot, heat one inch of oil to 350°. Fry 4 balls at a time, turning them once in the oil, until they are golden brown. It takes about 5 minutes, and then you can drain them on a wire rack. Make sure the oil heats up to 350° before you fry each batch.
Serve plain or with marinara sauce.