In a medium bowl, add the eggs and whisk until no streaks remain.
Add the schmaltz, the chicken broth, 3 tablespoons of the dill, 1 1/2 teaspoons of the salt, and 3/4 teaspoon of the black pepper to the eggs and whisk vigorously to combine.
Add the matzo meal to the egg mixture and whisk to combine.
Cover and transfer the matzo dough mixture to the refrigerator and let it chill for at least 35 minutes and up to 2 days. This is critical to let the matzo meal hydrate.
Move an oven rack to the top third position and preheat the oven to 450 degrees F.
Line an 18x26-inch baking sheet with parchment paper.
On a large platter or clean rimmed baking sheet, add the chicken legs and evenly sprinkle them with 1 1/2 teaspoons of the salt.
Let the chicken sit at room temperature for at least 20 minutes.
On the prepared baking sheet, in a single layer, add the chicken wings and bake until golden-brown and the chicken reaches an internal temperature of 165 degrees F and the juices run clear, about 45-55 minutes.
In a large pot, add the baked chicken wings and any accumulated juices from the baking sheet.
Place the pot on the stovetop over medium-high heat.
Add the onions, the celery, the parsnips, the 1 chopped carrot, the parsley, the black peppercorns, and 4 quarts of the water to the chicken wings in the pot and bring the mixture to a simmer while stirring frequently.
Reduce the stovetop heat to medium-low if necessary to maintain a simmer, and cook until the stock is slightly reduced, about 40-50 minutes.
Add the seasoned chicken legs to the pot and simmer until the chicken legs are very tender and the meat reaches an internal temperature of 165 degrees F and the juices run clear, about 40-50 minutes.
In a separate medium pot, add the remaining water and bring to a boil.
Add the remaining 3 tablespoons of the salt to the boiling water.
Using damp hands, divide the chilled matzo dough mixture into 16 pieces and roll each piece into balls about 1 1/2-inches in diameter. Make sure to be gentle with the matzo dough, as too much pressure will result in tough matzo balls.
Using a slotted spoon, carefully lower the matzo balls into the salted boiling water. You may need to do this in batches to avoid crowding.
Reduce the stovetop heat to low and cover the pot.
Gently simmer the matzo balls, while checking them occasionally and adjusting the stovetop heat if boiling too rapidly, until the matzo balls are puffed and light in color, about 30-40 minutes. Do not remove any sooner or the ball will be dense and undercooked in the middle.
Transfer the cooked matzo balls from the stovetop to a bowl and let them rest while the soup finishes.
Repeat as needed until all of the matzo dough is used.
Transfer the cooked chicken legs to a large platter and let them sit until cool enough to handle.
With a fine-mesh sieve into a separate medium pot, strain the stock, then discard all the solids including the chicken wings (the wings could be used for another purpose).
When the chicken legs are cool enough to handle, remove the skin and the bones from the meat and tear the meat into bite-sized pieces.
Add the chicken pieces to the strained stock.
Place the pot with the chicken and stock on the stovetop over medium heat.
Add the remaining thinly sliced carrots to the strained stock mixture and cook until the carrots are just tender, about 4 minutes.
Taste the soup mixture and season it with the extra salt, as needed.
In eight serving bowls, using a slotted spoon, place 2 of the matzo balls into each bowl and ladle the soup mixture over the top.
Garnish the soup with the remaining chopped dill and the remaining black pepper and serve.