In a medium bowl, whisk egg and water together until well blended. Add turkey, breadcrumbs, cheese, parsley, garlic, salt, and pepper. Using moistened hands, shape turkey mixture into 1-¼-inch meatballs. Place on a baking sheet; chill 30 minutes. In a large pot, bring chicken stock to a boil. Reduce heat to medium and add carrots and turkey meatballs; simmer uncovered 8 minutes. Add pasta and cook until al dente (slightly undercooked), about 3 minutes. Stir in Swiss chard and simmer until tender, about 3 to 5 minutes longer. Season soup to taste with salt and pepper. Divide mozzarella into six serving bowls and ladle soup into bowls. Garnish with additional Parmesan cheese.