For the stock:
Place all of the ingredients, except salt, into a large stock pot with enough water to cover chicken, and bring to a boil. 

Reduce the heat to medium-low, skim off any foam, add salt, and simmer, partially covered, for about 45 to 50 minutes until the chicken is fully cooked (tender and falling off the bone).

Carefully remove the chicken from the broth and transfer to a bowl. Continue cooking the broth at a soft boil for 20 minutes or up to 2 hours to extract as much flavor as possible. Turn off the heat and let cool.

Remove the skin and bones from the chicken. Shred the chicken and reserve for later.

Strain broth through a fine sieve into a separate container, discarding solids.

For the matzoh balls:
In a large bowl, combine the matzoh meal, salt, pepper, baking powder and baking soda.

In a medium bowl, whisk the 2 whole eggs with the 3 yolks, schmaltz and onion. 

In a separate bowl, beat the 3 egg whites by hand or with an electric 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.

Press a sheet of plastic wrap directly onto the surface of the batter and refrigerate for about 20 minutes or overnight, until firm.

To assemble & serve:
Line a baking sheet with plastic wrap. In a small bowl, combine the vegetable oil with 1 tablespoon of water. Scoop 1-tablespoon-size mound of the matzoh batter onto the baking sheet. Using the oil-and-water mixture to keep your hands moist, roll each scoop of batter into a ball, handling them as gently as possible.

Return the chicken stock to a clean pot and heat to a simmer. Lower the matzoh balls as gently as possible into the broth and cook over medium heat, turning the matzoh balls a few times, until they are plump and cooked through, about 25 minutes. 

Stir the shredded chicken into the soup and cook just until the meat is warmed through, about 2 minutes. Taste and season with salt and pepper. Serve immediately, with garnishes on the side for people to add as preferred.