Heat 2 tablespoons of extra virgin olive oil in a large skillet over medium heat. Add the finely chopped onions, carrots, celery, and 2 tablespoons chopped fresh parsley, stirring to coat with oil. Reduce the heat to low, cover the skillet, and cook until the vegetables are soft, about 15 to 20 minutes. Remove the cover, add 1 minced garlic clove, then increase the heat to medium-high and cook for 30 seconds. Stir in the canned tomatoes (breaking them up if using whole), tomato paste, dried basil, and season with salt and black pepper. Bring the mixture to a gentle simmer, then reduce the heat to low and cook uncovered until the sauce thickens, about 15 to 30 minutes. If a smoother texture is desired, purée the sauce in a blender or use a food mill. Return the sauce to the skillet and adjust seasoning, if needed. I like tasting at this stage and adjusting the salt or adding a pinch more basil for a brighter flavor.
Heat the milk in a small pot until it becomes steamy, then remove from the heat. Tear the slices of white bread into small pieces and add them to the warm milk, allowing the bread to soak until it starts to dissolve. Mash the mixture with a fork until it forms a loose paste, then turn it out onto a plate to cool. This step makes the meatballs wonderfully tender.
In a large bowl, combine the ground beef, ground pork, ricotta cheese, grated Parmesan or Romano cheese, eggs, 3 minced garlic cloves, 1 tablespoon chopped fresh parsley, dried oregano or marjoram, 1 teaspoon salt, freshly ground black pepper, and the bread-milk mixture from Step 2. Gently mix everything by hand until just combined, being careful not to overwork the mixture. Wet your hands and form the mixture into meatballs about 2 to 3 inches across (or preferred size). Roll each meatball in flour for a light coating and set on a baking sheet. You may need to rinse and re-wet your hands a few times during this process. I find that handling the mixture as little as possible keeps the meatballs extra tender.
Heat 1/3 cup extra virgin olive oil in a large skillet over medium-high heat. Working in batches if necessary, brown the formed and floured meatballs from Step 3 on at least two sides. Do not worry if the meatballs are not cooked through, as they will finish cooking in the sauce. Remove the browned meatballs and set aside while you finish the rest. For the best flavor, I like to let the meatballs get a rich golden crust on the outside during this step.
Transfer the browned meatballs from Step 4 into the thickened tomato sauce from Step 1, turning each meatball to coat it well in the sauce. Cover the pot and simmer gently for 15 to 20 minutes, allowing the meatballs to cook through and absorb the sauce’s flavors. Serve hot, optionally garnished with a sprinkle of chopped parsley and extra sauce over pasta or with crusty bread. I like to finish the dish with a few more shavings of Parmesan for extra richness.