Use a pasta pot or stock pot that holds 16 cups of water with room to spare for chicken and vegetables. Add all the vegetables and chicken into the pot. I usually put the chicken in first and then add all the vegetables and pour in the water.
Place the stove on medium to medium high heat to bring the mixture to a boil, then turn it down to simmer. You’ll want to see a bit of bubbling here and there but not an active boil anymore.
As it works its way to a boil, you’ll see the water get cloudy and some scum accumulate at the top. Skim this off occasionally – a large spoon will do the trick.
At about an hour in, check the chicken. If it’s completely cooked (leg will jiggle easily), take it out of the pot carefully and place on a large plate to cool off a bit. This is a good time to walk your dog. The goal is for the chicken to cool enough that you can shred the meat off the carcass.
While the chicken cools down a bit, let the rest of the stock continue to simmer on the stovetop.
When you return from walking the dog, the chicken will have cooled down enough to take all the meat off the bones. Put the bones back in the pot and let it continue to cook – about another hour. Reserve the chicken meat in a container in the refrigerator.
When the soup stock is done, let it cool down and then strain it through a fine mesh colander, lined with cheesecloth if necessary. Discard the solids and store the gorgeous golden-brown stock in containers in the fridge or freezer for later use. Do let it cool down a bit before closing the containers. Even if freezing, I would refrigerate to get the fat off the top if you want to use it for the matzo balls.
Note that it is optional to remove the chicken and take the meat off the carcass. Many people just boil the chicken for a few hours and discard the meat. I prefer to kill two birds with one stone, so to speak, and save the chicken for use in the soup or whatever else you might need shredded chicken for.