Use your fingers or kitchen shears to pull the chicken carcass into a few pieces that will fit snugly in the pot.
Put the chicken bones in a pot and cover them with water by about an inch. Simmer on very low heat for 2-6 hours. You should just see a few bubbles here and there, a little movement in the liquid, and a little bit of steam over the pot. Add more water if the bones start to become exposed. The ideal temperature is between 180° and 190°.
Peel and roughly chop all of the veggies.
Skim off any foam or film that has floated to the top of the stock.
Add the vegetables and herbs. Add more water if necessary to cover. Simmer for another hour or two at the same heat.
Strain the stock to separate out the solid pieces. Discard the solids.
Let the stock cool, then separate into portion-sized containers. Refrigerate stock for up to a week or freeze it indefinitely.