Heat 1/4 cup olive oil in a 3-quart saucepan over medium heat. Crumble in the meat and add the onion. Cook, stirring often, until the water given off by the meat is evaporated and the meat and onion begin to brown, about 10 minutes. Season the beef and onion lightly with salt. Stir in the carrots and celery and continue cooking until the vegetables are tender, about 10 minutes. Stir in the tomatoes, tomato paste, red pepper, and salt to taste. Adjust the heat to simmering and continue cooking, stirring occasionally, until the sauce is thickened, about 30 minutes. If the sauce starts sticking to the pan at any time during cooking, stir in a few tablespoons of water. Stir in the peas and cook until they are very tender, about 10 minutes for frozen peas and 20 minutes for fresh peas. The finished ragù should be dense and reduced. Remove and cool to room temperature. While the ragù is cooling, make the rice Bring the stock or water and 2 tablespoons olive oil to a boil in a 3-quart saucepan. Stir in the rice, return the water to boil, then adjust the heat to simmering. Cook the rice, uncovered, until al dente—tender but firm—about 12 minutes. Drain the rice and spread out on a tray to cool to room temperature. When the rice is cool, scrape it into a mixing bowl and beat in the 4 eggs and the grated cheese. Take a handful (about 1/3 cup) of the cooled rice mixture and shape it into a small ball in the palm of your hand. Make a well in the center of the ball and drop in 1 tablespoon of the ragù. Work the rice so that it completely encloses the ragù, and re-form the rice into a smooth ball. Continue forming arancine with the remaining rice and ragù. Whisk the 2 eggs in a mixing bowl. Spread the flour on one plate and the bread crumbs on another, in an even layer. Dredge a few of the rice balls in flour to coat all sides. Tap off excess flour. Roll the rice balls in the beaten egg to coat, allowing any excess egg to drip back into the bowl. Finally, roll the rice balls in the bread crumbs, pressing lightly to coat evenly with the crumbs. Remove to a clean baking sheet. Repeat with the remaining rice balls. If you’d like to serve the rice balls hot, heat the oven to 200° F or to the lowest setting. Line a baking sheet with a double thickness of paper towels. Pour the vegetable oil and olive oil into a deep skillet. Insert a deep-frying thermometer in the oil and heat the oil over medium heat to 375° F. (If you are working without a thermometer, test the temperature as directed below.) Once the oil reaches temperature, adjust the heat under the pot to maintain a steady temperature. If you’re not working with a thermometer, test the temperature of the oil by dipping a rice ball in the oil. It should give off a lively but steady sizzle. If nothing happens, the oil isn’t hot enough; if the oil around the bread-crumb coating boils and sputters, the oil is too hot. Adjust the heat accordingly. When the oil comes to temperature, carefully slip about a third of the rice balls into the oil. Fry, turning as necessary with tongs or a slotted spoon, until golden brown and crisp on all sides, about 4 minutes. Remove to the paper-towel-lined baking sheet, keeping them hot in the oven if you like. Fry the remaining rice balls. The arancine can be served hot or at room temperature.
