Place chicken into a large pot and cover with water; bring to a boil. Reduce heat to low and simmer until chicken meat is very tender and falling off the bones, about 1 1/2 hours. Remove chicken from pot and pick chicken meat from bones when cool enough to handle, discarding skin and bones. Shred chicken meat.
Bring chicken broth to a boil in a large soup pot; stir shredded chicken meat, carrots, celery, turnip, parsnips, leek, onion, and dill into broth. Turn heat to low and simmer until vegetables are tender, about 20 minutes.
Stir matzo ball mix, eggs, vegetable oil, and 1/2 cup water together in a bowl. Form mixture into 1-inch balls using wet hands to prevent sticking. Return broth to a boil and gently drop matzo balls into the boiling soup. Reduce heat again and simmer soup until matzo balls have increased in size and are cooked through, about 20 more minutes.