How to Make It
Preheat the oven to 425°. Spread the chicken wings in a large roasting pan and roast for about 30 minutes, until golden brown.
Meanwhile, in a large stockpot, heat the vegetable oil. Add the onions, cover and cook over moderately high heat, stirring occasionally, until softened, about 5 minutes. Uncover and cook, stirring frequently, until the onions are lightly browned, about 5 minutes longer. Add the chicken backs, necks and feet, the diced carrots, celery, bay leaves, garlic, thyme, rosemary, peppercorns and 8 quarts of water and bring to a light boil, skimming any scum that rises to the surface.
Add the roasted chicken wings to the stockpot. Place the roasting pan over 2 burners, add the vinegar and 2 cups of water and simmer over high heat, scraping up any browned bits. Add the liquid to the pot and simmer gently over moderate heat for 2 1/2 hours. Carefully strain the stock into another large pot and discard the solids, including the wings. You should have about 8 quarts. Let the stock cool, then spoon off all of the chicken fat and reserve it for the matzo balls. Season the chicken stock lightly with salt.
In a medium bowl, whisk the eggs with the chicken fat, club soda, scallions, rosemary and salt. Whisk in the matzo meal. Refrigerate for 1 hour, until slightly firm.
Bring a small saucepan of salted water to a boil. Add the sliced carrot and leeks and cook over high heat until tender, about 7 minutes; drain. In a saucepan, bring 2 cups of the stock to a simmer. Add the diced chicken breast and thighs and cook over moderately low heat until just white throughout, about 6 minutes. Transfer the chicken to a plate; reserve the broth for another use (see Note).
Bring a large saucepan of the chicken stock to a simmer. Scoop rounded tablespoons of the matzo mixture into balls (about 2 dozen balls) and add them to the simmering stock. Cover and cook over very low heat until the matzo balls are tender, fluffed and cooked through, about 25 minutes. Using a slotted spoon, transfer the matzo balls to a clean pot. Add the chicken, carrots and leeks and 8 cups of the stock and bring to a simmer. Ladle the soup and matzo balls into bowls and serve.