If using a whole chicken, cut it into 8 pieces (2 thighs, 2 breasts, 2 wings, 2 legs). Place the chicken pieces or whole chicken legs in an 8-quart or larger stockpot.
Prepare the vegetables (no need to peel them first) and add them to the stockpot as you go: Quarter the 2 yellow onions and halve the garlic head horizontally. Coarsely chop the celery, parsnips, and 2 of the carrots.
Add the parsley, kosher salt, and black peppercorns to the pot. Pour in the cold water (if your pot isn’t large enough, you can add the remaining water as the stock reduces).
Bring the mixture to a lively simmer over high heat. Once simmering, reduce the heat to low to maintain a gentle simmer. Let it simmer uncovered for 1 hour. Meanwhile, peel and slice the remaining 2 carrots on a slight diagonal.
Using tongs, transfer the chicken breasts and thighs (or 2 whole legs) to a plate. Continue to simmer the broth for about 1 more hour until flavorful. Once cool enough to handle, shred the meat from the removed pieces and discard the skin and bones.
Fit a colander over a large bowl and pour the broth through it, discarding the solids. Return the broth to the stockpot. Taste and season with kosher salt and black pepper as needed. You can cool and refrigerate the broth at this point, along with the shredded chicken and sliced carrots.
In a medium bowl, whisk the 5 eggs until broken up. Finely chop the fresh dill and add it to the bowl. Mix in the matzo meal, melted schmaltz or butter, kosher salt, and black pepper. For lighter matzo balls, add the seltzer or club soda and baking powder. For denser matzo balls, use chicken broth instead and skip the baking powder. Whisk to combine, cover, and refrigerate for at least 2 hours or overnight.
Bring a large pot of heavily salted water to a boil. Line a baking sheet with parchment paper. Scoop the matzo mixture into 1 1/2-tablespoon portions and roll into balls (you can oil or dampen your hands if the mixture is sticky) and place them on the baking sheet.
Add the matzo balls to the boiling water and simmer until they float and are fully cooked, about 10 minutes. Meanwhile, bring the broth to a simmer over medium heat and add the reserved sliced carrots and shredded chicken, if using. Simmer until the carrots are slightly softened, about 10 minutes.
Using a slotted spoon, transfer the matzo balls into the soup and simmer for an additional 5 minutes. Serve the soup garnished with fresh dill and freshly ground black pepper if desired.
Notes
For a heartier soup, feel free to add more vegetables like potatoes or turnips. You can make the broth a day ahead and let the flavors develop overnight.