Combine the ground beef, breadcrumbs, Parmesan, egg, minced garlic, Italian seasoning, kosher salt, and black pepper in a bowl. Mix only until the ingredients are combined, taking care not to compact the meat.

Shape the mixture into approximately 30 small meatballs, using about 2 teaspoons of mixture for each one. Aim for meatballs close to 1 inch in diameter so they cook evenly.

Heat the olive oil in a large soup pot or Dutch oven over medium heat. Brown the meatballs in batches for 4 to 6 minutes, turning them to brown several sides, then transfer them to a plate; they do not need to be fully cooked yet.

Add the diced onion, carrots, and celery to the same pot. Cook for 4 to 5 minutes, until the vegetables begin to soften, then add the minced garlic and cook for 30 seconds.

Pour in the broth, undrained diced tomatoes, tomato sauce, Italian seasoning, kosher salt, and black pepper. Bring the soup to a boil, scraping up any flavorful browned bits from the bottom of the pot.

Return the browned meatballs to the pot. Reduce the heat, partially cover the pot, and simmer for about 15 minutes, or until the meatballs are cooked through.

Add the broken spaghetti and simmer for 8 to 10 minutes, stirring occasionally, until the pasta is tender. Taste the broth and adjust the seasoning with more kosher salt or black pepper as needed.

Ladle the soup into bowls and finish with chopped fresh parsley, if using, and grated Parmesan. Serve promptly while the spaghetti is tender and the broth is plentiful.