Add the meat to a large bowl, spread it out and season with salt, to taste. Add the remaining ingredients leaving the bread crumbs until last. Mix until all the ingredients are combined. Create a small patty and test it in a small saute pan in 1 tablespoon of oil. When browned on both sides, taste and reseason the meat mixture, if needed. Roll the mixture into about 20 to 25 or so balls that are about 2 inches in diameter.
Heat the vegetable oil in a large skillet over high heat. When the oil begins to smoke lightly, shut off the heat to avoid splattering, add the meatballs in a single layer and spread them somewhat apart so they have a chance to brown instead of steam. You may need to do this in batches. Brown the meatballs, over high heat, turning them so they brown all around. Treat them like hamburgers and cook them until they are medium-rare, about 3 to 5 minutes. Touch them to make sure they are still tender in the center. Use a slotted spoon or spatula to remove them from the pan to a tray and cook off the remaining balls, if needed.
Add the meatballs to the hot tomato sauce and allow them to bubble slightly and simmer over very low heat for a few minutes. Shut off the heat and allow the mixture to rest for a few additional minutes before serving.
Heat the olive oil in a large skillet over medium heat. Add the onions, the garlic, red pepper flakes, and sugar and season with salt, to taste. Stir in the carrots and reseason with salt. Cook for about 2 minutes, and then add canned tomatoes. Use a wooden spoon to break up some of the whole tomatoes and cook over medium heat, stirring from time to time, about 5 to 10 minutes. Add half of the water to prevent the veggies from getting too dry and continue cooking another 10 minutes. Taste for seasoning. The tomatoes should be fairly broken down and the flavors coming together. Add remaining water, as needed, and cook for an additional 10 minutes. The sauce cooks about as long as it takes to make the meatballs from start to finish, about 45 minutes. Stir in the basil leaves and season with Parmesan.