Heat 1/4 cup olive oil in a stockpot over medium heat. Cook neck bones, country-style pork rib, beef chuck, beef rib, and lamb shank in hot oil, turning regularly, until browned on all sides, 7 to 10 minutes. Remove all the meat to a large bowl.
Pour another 1/4 cup olive oil into the stockpot and heat over medium heat; stir a pinch of basil, a pinch of mint, and a pinch of red pepper flakes into the oil. Cook and stir onions in hot oil until translucent, about 5 minutes. Add garlic; cook and stir 1 minute more. Stir the tomato paste into the onion mixture until incorporated. Pour crushed tomatoes and water into the mixture and bring to a low boil. Add an additional 2 pinches each of basil, mint, and red pepper flakes. Season with salt and black pepper.
Return the browned meats to the tomato mixture; bring to a simmer and cook, stirring every 15 minutes, for 2 1/2 hours. Remove the neck bones and discard.
Heat 2 tablespoons olive oil in a skillet over medium heat. Crumble Italian sausage into the skillet; cook and stir until crumbly, evenly browned, and no longer pink, 5 to 7 minutes. Drain and discard any excess grease. Stir sausage into the tomato sauce with sugar. Return the sauce to a simmer and cook another 1 1/2 hours.
Mix ground beef, ground pork, eggs, milk, Italian bread crumbs, parsley, Parmesan cheese, and garlic with your hands in a large bowl until evenly mixed; shape into 2-inch balls.
Heat 1 1/2 teaspoons in a skillet over medium heat. Cook the meat balls in hot oil until evenly browned, 5 to 7 minutes.
Add meatballs to the tomato sauce. Cook at a simmer for 1 hour, adding water as needed to keep sauce from becoming too thick. Remove stockpot from heat; let sit 2 to 3 minutes. Skim and discard any fat from the top of the sauce.
