Combine the ground beef, breadcrumbs, egg, Italian seasoning, salt, and pepper in a mixing bowl. Mix gently until just combined, then roll the mixture into small meatballs about one inch across.
Place the meatballs into the slow cooker. Add the chicken broth, carrots, celery, onion, and garlic.
Cover and cook on low for 6 to 7 hours until the meatballs are cooked through and the vegetables are tender.
Stir in the small pasta during the last 30 minutes of cooking so it softens without becoming mushy.
Turn off the heat and stir in the fresh spinach until it wilts.
Ladle the soup into bowls and top each serving with grated Parmesan cheese.