For the Meatballs: In a large bowl, use a clean hand to thoroughly mix together the ground meat with the garlic, eggs, breadcrumbs, parsley, Parmigiano-Reggiano, oregano, salt, and a generous grinding of black pepper until meat develops a slightly sticky texture. Set aside.
For the Soup: In a large soup pot or Dutch oven, heat olive oil over medium heat until shimmering. Add carrots, celery, and garlic, and cook, stirring often, until vegetables have softened but not browned, about 6 minutes.
Add chicken stock and bring to a simmer. Add leafy greens and cook until wilted and tender.
Working quickly with clean hands, roll meatball mixture into 1-inch balls and drop into simmering soup. Cook until meatballs are all cooked through, about 5 minutes. Season with salt and pepper.
When ready to serve, add about ½ cup cooked pasta to each serving bowl, then ladle soup, meatballs, and vegetables on top. Top with grated cheese and serve.
