In a large pot, bring 2 quarts chicken stock to a simmer. Add the chicken and return just to a simmer. Cover the chicken with a small heatproof plate to keep it submerged (add more stock if needed). Cover the pot and reduce the heat; gently simmer until the chicken is cooked through, about 1 1/2 hours.
Remove the chicken and let cool slightly, then roughly cut up the meat and set aside; discard the skin and bones. Strain the stock into another large pot.
Skim the fat off the surface of the stock using a spoon.
Make the batter.
Meanwhile, in a large bowl, whisk the matzo meal, granulated garlic, baking powder, baking soda, 2 1/2 teaspoons salt and a few grinds of pepper. In a medium bowl, whisk the 2 whole eggs with the 3 yolks, schmaltz and minced onion. In a separate clean bowl, beat the 3 egg whites with a hand mixer until stiff peaks form. Stir the schmaltz mixture into the dry ingredients, then stir in one-third of the beaten egg whites until incorporated. Gently fold in the remaining egg whites until no streaks remain.
Form the matzo balls.
Line a baking sheet with plastic wrap. In a small bowl, combine the vegetable oil with 1 tablespoon water. Using the oil-and-water mixture to keep your hands moist, roll scoops of batter (about 2 tablespoons each) into balls, handling them as gently as possible (makes 8 to 12). Arrange on the baking sheet and refrigerate 20 minutes so the outsides dry slightly.
Make the soup.
Tie the dill and parsley sprigs together with kitchen string. Return the chicken stock to a simmer. Add the carrot, celery, diced onion, rutabaga and herb bundle; season with salt and pepper. Return to a simmer and add the matzo 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 chicken into the soup and cook until just warmed through. Remove the herb bundle and season the soup with salt and pepper.
If you're making this soup for Passover, use ingredients specifically marked kosher for Passover.