Place the ground meat, egg, Parmesan, bread crumbs, 2 teaspoons oregano, 1/2 teaspoon paprika, 1/2 cup parsley, and 2 to 3 minced garlic cloves in a large bowl; add a drizzle of olive oil, salt, and black pepper, then mix gently until combined. The mixture should look evenly seasoned but still feel loose and soft rather than compacted.

Roll the mixture into meatballs using about 1 1/2 tablespoons for each, then arrange them on a lightly oiled large sheet pan and broil for 5 to 8 minutes. Look for sizzling edges and a light golden-brown surface, while the centers remain unfinished.

Heat 2 tablespoons extra virgin olive oil in a large pot over medium-high heat, then cook the onion, carrots, celery, bell pepper, and remaining 1 to 2 garlic cloves for 5 minutes, stirring occasionally. The onions should turn translucent and the vegetables should smell sweet and aromatic without browned garlic.

Add the broth, tomatoes with their juices, remaining 1 teaspoon oregano, and remaining 1/2 teaspoon paprika; season with salt and black pepper, then simmer over medium to medium-high heat for about 10 minutes. Keep the pot at a steady bubble until the broth looks deeper in color and slightly more glossy.

Slide in the broiled meatballs and uncooked pasta, then continue cooking for about 10 minutes. The pasta should be tender with a little bite, and the meatballs should yield gently to spoon pressure without breaking apart.

Take the pot off the heat and stir in the remaining 1/2 cup parsley before ladling into bowls with Parmesan and red pepper flakes, if using. The fresh parsley should stay bright green and lift the rich tomato aroma.