Place the chicken in a large pot so it fits snugly and fill the pot with enough water to cover the chicken by 1 inch. Bring to a boil over high heat, then reduce the heat to maintain a simmer. Cook for 15 minutes, using a ladle to skim the foam that rises to the surface. Add the garlic, bay leaves, peppercorns and 1 tablespoon plus 1 teaspoon salt and continue cooking for another 30 minutes. Skim any fat that rises to the surface and reserve. You should have about 1/4 cup -- if necessary, add vegetable oil to equal 1/4 cup and set aside to cool slightly. Add the celery, carrots, onion and parsley sprigs to the pot and continue to cook until the vegetables are soft and the chicken is tender but not falling apart, about 45 minutes.
Meanwhile, whisk together the eggs and chicken fat in a medium bowl, then stir in the seltzer, chopped dill, matzo meal and 1 1/2 teaspoons salt until combined; cover and refrigerate. Chill until the mixture is firm, at least 30 minutes and up to 2 hours.
When the chicken is cooked, carefully remove it from the broth with a slotted spoon and place on a rimmed baking sheet along with the vegetables; let cool. Return the broth to a simmer. Scoop heaping tablespoons of the matzo mixture, using your hands to gently roll them into balls (wet your hands to prevent sticking). Carefully drop the matzo balls into the simmering broth. Cover and simmer on low until the matzo balls are completely cooked through and fluffy, about 1 hour.
While the matzo balls cook, pick the meat from the chicken and roughly chop it, discarding the skin and bones. Slice the celery and carrots into 1/4-inch-thick pieces and discard the onion and aromatics.
When ready to serve, return the chicken, celery and carrots to the broth. Increase the heat to high to bring the soup to a simmer, then remove from the heat. Gently ladle the soup in to serving bowls, being careful not to break up the matzo balls. Top with fresh chopped dill and serve.
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