Prepare a large baking sheet lined with parchment paper and have a bowl of water nearby to assist with rolling the rice balls.
Grab a 1/2 cup of rice with a measuring cup or a similar-sized ice cream scooper and place it into one hand.
Make an indentation into the center of the ball and then add 1-2 teaspoons of the ragu and 1-2 small cubes of caciocavallo cheese. Add a touch more rice to close it up then roll it into a uniform ball. If needed, dip hands in water to add a bit of moisture which can assist in the balls staying together. Place all of the rice balls onto the parchment-lined baking sheet.
In a bowl, mix the flour, water, salt, and eggs together until smooth. Also prepare a plate with the breadcrumbs.
Dip a rice ball into the batter then let the excess drip off before rolling in the breadcrumbs. Repeat the rolling process for all of the rice balls then place onto the parchment-lined sheet. Place the formed rice balls into the fridge while heating the oil.
Heat a large heavy and deep Dutch oven with 3 inches of vegetable oil to 370f. Make sure to not pour the oil too high, since the height will increase when frying the arancini.
Once the oil is at the proper temp, begin frying the arancini. Depending on the size of your pot, fry no more than 3 at a time. Fry until they are equally browned on all sides (about 4-5 minutes total). Remove the arancini with a slotted spoon or spider to a paper towel lined baking sheet. Eat right away!