Mix the eggs and meat thoroughly to evenly distribute. Add the Pecorino Romano cheese, breadcrumbs, chopped parsley, garlic, salt and pepper.
Mix thoroughly together by hand in order to combine all the ingredients.
Then, start making your balls. Remember to compress the ball with your hands so they are very compact and will not crack. This is important because a ball with any cracks, or not compact enough, may break during the shallow frying stage.
Now, heat the olive oil in a large frying pan and fry the meatballs over medium heat until golden brown. If the meatballs stick to the bottom of the pan, you may use a non-stick frying pan or you can fry the meatballs two at a time and keep rolling then in the hot oil.
Place the meatballs on a new clean plate or a tray.
Do not discard the olive oil but filter it into a small bowl in order to remove the residue created when frying the meatballs. Return filtered oil to the pan, heat, and add the chopped onion and garlic. Sweat off for about 2 minutes.
Add the tomato puree and cook on medium heat for about 3 minutes, stirring constantly.
Add 150 ml (5 fl oz or 5/8 cup) of hot water to the frying pan and cook for one minute, stirring. The gravy is done, so remove the pan from the heat and set aside.
In a large saucepan (a casserole would do the job), add the chopped tomatoes and cook on medium heat for 5 minutes.
Then, add the gravy (tomato puree mixture) from the frying pan. Add the reserved browned meatballs. Cover with hot water. In my case, I used a very large saucepan so to cover the meatballs, I needed 2 ½ cups of water. A slightly smaller saucepan would probably require less water, I would say 1 2/3 cups.
Add the chopped parsley, a pinch each of salt and pepper. Mix thoroughly, but carefully, with a wooden spoon, so as to not break the meatballs. Now, cook on medium heat for about 15 minutes.
Cover (leaving one side of the lid slightly open) and simmer for about 2 ½ hours, stirring every 20 minutes to prevent sticking on bottom of the pan. Also, when doing that, turn the meatballs so the part that was face up will get the next 20 minutes cooking facing the bottom and keep going like that until the end. After 1 ½ hours simmering, check for seasoning; usually you need to add some salt.
Serve the meatballs over al dente pasta and use the red sauce from the pan to coat the pasta. Finally, sprinkle with freshly grated Pecorino Romano cheese (alternatively Parmesan cheese).