Add 1.2 liters of water to a saucepan and heat over high heat. When it boils, add 1 tablespoon of salt and rice. Simmer for 15 minutes, stirring often, until the rice is cooked, and the water is completely absorbed.
In a bowl, add cooked rice and butter. Stir to create a creamy and uniform mixture.
Spread the rice onto a large baking sheet in a layer of about 1 cm and cover it with a transparent film. Let the rice cool completely at room temperature (this will take a couple of hours).
Take a couple of tablespoons of rice with your hand. Hold your hand in the shape of a cup and use your other hand to flatten the rice until it forms a cup-like shape. The thickness must be at least 1.5 cm, otherwise the filling will flow out. To help you with this step, have a bowl filled with water nearby, to moist your hands if you needed.
Add a teaspoon of tomato sauce, a piece of mozzarella and 3–4 pieces of ham to the center of the rice. Then add a spoonful of rice on top to cover the ball. Shape Arancino with your hands until it turns into a ball. Repeat with the remaining rice.
In a large bowl, add the sifted flour, a teaspoon of salt, and 300 ml of water. Beat well with a whisk so that no lumps form.
Add breadcrumbs to a plate. Dip the Arancini one by one into the batter, being careful to coat them completely. Then roll them in breadcrumbs, reshaping them with your hands. Set aside until all the rice balls are coated.
Heat the oil in a saucepan and bring it to 170 °C. Check the temperature with a cooking thermometer. Deep-fry one Arancino at a time, or two at a time. Flip Arancini often, so they don't brown too much on one side. When the rice balls are golden brown, remove them from the oil with a slotted spoon and place them onto a baking sheet lined with absorbent paper.
Your Italian rice balls are ready. Spread a serving dish with a little tomato sauce and arrange 2-3 still hot arancini on top. Garnish with some fresh basil leaves if desired. Enjoy them immediately!