In a large pot, bring the chicken stock to a simmer. Add the chicken and return the stock just to a simmer. Cover the chicken with a small plate to keep it submerged and cover the pot. Reduce the heat to maintain a very low simmer; simmer until the chicken is cooked through, about 1 1/2 hours. Remove the chicken and let cool slightly, then shred the meat; discard the skin and bones. Strain the soup into a heatproof bowl. Skim off the fat and return the soup to the pot.
Make the matzo balls following directions on package while chicken is cooking.
Line a baking sheet with plastic wrap. In a small bowl, combine the vegetable oil with 1 tablespoon of water.
Scoop 1-tablespoon-size mounds of the matzo batter onto the baking sheet. Using the oil-and-water mixture to
keep your hands moist, roll each scoop of batter into a ball, handling them as gently as possible.
Return the chicken soup to a simmer. Add the carrot, celery, onion, rutabaga, dill and parsley and season with a big pinch each of salt and pepper. Add the matzoh balls. Cover and cook over moderate heat, turning the matzo balls a few times, until they are plump and cooked through, about 25 minutes. Stir the shredded chicken into the soup and cook just until the meat is warmed through, about 2 minutes. Remove the herb sprigs. Season the soup with salt and pepper and serve immediately.
Make Ahead: The soup can be prepared through Step 3 and refrigerated overnight. Cover the matzo balls and shredded chicken with plastic wrap before refrigerating.