Melt butter in a medium pot over medium-high heat. Add onion and 1/2 tsp. salt. Cook, stirring, until soft, about 2 minutes. Add turnips, parsnips, broth, and 1 cup water. Bring to a boil, then reduce heat to medium-low and simmer until vegetables are tender, 10 to 15 minutes.
Meanwhile, combine pork, 1 tbsp. chives, parsley, cheese, 1/2 tsp. salt, and 1/4 tsp. pepper in a small bowl. Mix gently to combine. Form into 36 balls, about 1 tsp. each, and set aside.
Whirl soup in a blender until smooth (in batches if necessary). Return to pot and keep warm over low heat.
In a wide, shallow pan, bring at least 2 in. of water to a boil and add remaining 1 tsp. salt. Add meatballs and cook until they float to the surface and are cooked through (test one to be sure), about 4 minutes. Transfer to a warm plate with a slotted spoon.
Divide soup among 6 bowls. Top each bowl with 6 meatballs and sprinkle with additional chives and pepper. Serve hot.
Note: Nutritional analysis is per serving.