Prepare the Bouquet garni by wrapping fresh thyme sprigs, parsley stems, bay leaf, black peppercorns, and cloves in a piece of cheesecloth. Secure the bundle by tying it with kitchen twine or by hand, creating a pouch.If you don't have cheesecloth, you can simply add the fresh thyme sprigs, parsley stems, bay leaf, black peppercorns, and cloves directly into the stock pot. You can strain them out later using a fine mesh strainer before serving. Roughly chop the carrot, leek, celery, and onion, and gently crush the unpeeled garlic cloves before placing them in a large stock pot or Dutch oven. Add the chicken carcass and bouquet garni (or all the ingredients for the bouquet garni directly) to the pot.
Add 10 cups of cold water into the pot and place it on the stove over medium heat. Once it's at the point of boiling, lower the heat to medium-low and simmer for 3 hours.
During the simmering process, use a fine mesh skimming tool to remove any impurities that form on the surface of the stock. This may need to be done multiple times, particularly at the beginning of the cooking process.
Ensure the ingredients remain submerged in water at all times by monitoring the water level during the cooking process.
After the simmering process is complete, use a colander or sieve lined with cheesecloth to strain the stock into a large container. Discard the remaining solid ingredients.If you do not have a cheesecloth for straining, you can use a fine mesh strainer or a slotted spoon to remove the solid ingredients from the stock. Be cautious to preserve as much of the liquid as possible during the straining process. Allow the stock to cool completely, taking care not to let it sit out for more than 2 hours to prevent contamination. For faster cooling, place the container in your sink, elevating it by putting it on top of something, and fill the sink with ice cubes and cold water.
Refrigerate the stock immediately once it has cooled. After a few hours or overnight, remove the layer of fat that has formed on the top.
You can store the stock in an airtight container in the refrigerator for up to 2 days, or transfer it into freezer bags and keep it in the freezer for up to 6 months.