Heat 2 tablespoons of the oil in a small saute pan over medium heat. Add the garlic and cook until soft, about 1 1/2 minutes.
Whisk together the eggs and parsley in a large bowl. Add the browned garlic, meats and cheese and gently mix together until combined. Begin adding bread crumbs 1 tablespoon at a time until the mixture just holds together. Season with salt and pepper, to taste. Cover and refrigerate for at least 30 minutes. Form the meat into 1 1/2-inch balls.
Heat the remaining 1 cup of oil in a large saute pan over high heat. Add the meatballs and fry until golden brown on all sides. Carefully remove to a baking sheet lined with paper towels.
Heat the oil in a medium saucepan over medium-high heat. Add the onion, garlic and chili flakes and cook until the onion is soft. Add the tomato paste and cook for 30 seconds.
Stir in the tomatoes and let simmer, uncovered, for about 10 minutes. Add the peppers and parsley sprigs, season with salt and pepper, to taste, bring to a boil and cook for 10 minutes, stirring occasionally. Add the meatballs, reduce the heat to medium and continue cooking until the meatballs have cooked through and the sauce has thickened, about 30 minutes longer.
Drain the spaghetti and return to the pot. Add some of the sauce and toss to coat for about 30 seconds. Stir the chopped basil into the tomato sauce and ladle some of the sauce over the spaghetti and meatballs. Grate lots of Parmesan over the top and garnish with fresh parsley sprigs.