In a medium pot, heat 2 tablespoons olive oil over medium heat.
Add 2 cups Arborio rice and stir for 1–2 minutes to lightly toast. Season with ½ tsp salt and ¼ tsp black pepper, then pour in 2 cups chicken stock or water.
Bring to a boil, cover, and reduce to a simmer. Cook for about 15 minutes, or until the liquid is absorbed and the rice is al dente. Do not stir while it cooks.
Remove the cooked rice from the heat and stir in ½ cup grated Pecorino Romano or Parmesan while the rice is still hot.
Spread the rice thinly onto a baking sheet to stop the cooking process and allow it to cool for about 15 minutes at room temperature. Then transfer the baking sheet to the refrigerator and chill until completely cold.
Place the cold rice in a large bowl and mix in the 3 eggs until fully combined.
With damp hands, take about 2 tablespoons of rice and flatten it slightly in your palm. Add a small spoonful of ragù, a few peas, and a cube of mozzarella. Cover with 2 more tablespoons of rice and shape into a ball or a cone, making sure the filling is completely enclosed. Repeat with the remaining rice and filling.
Arrange the shaped arancini on a tray and refrigerate for at least 30 minutes, or until firm.
Prepare a coating by whisking 3-4 eggs.
Dip each arancino into the eggs, then roll in 2 cups seasoned breadcrumbs until evenly covered. For a thicker crust, repeat the coating and breadcrumb step a second time. Return to the refrigerator for another 30–60 minutes.
Add enough vegetable oil to a deep pot to cover the arancini, and heat it to 350°F. Fry the arancini in batches for 3–5 minutes, turning as needed, until golden brown and crisp on all sides. The centers should be heated through and the cheese melted (an internal temperature of about 150 degrees Fahrenheit) .
Remove with a slotted spoon and drain on a paper towel-lined plate. Serve warm.