Place the chicken carcass in a large stockpot.
Add the chopped onion, shallot, celery, carrots, mushroom stems, thyme, bay leaf, and kosher salt to the pot.
Pour in enough water to cover all the ingredients.
Cover the pot with a lid and bring it to a boil over high heat. Then, reduce the heat to low and let it simmer for 3.5 to 4 hours, slightly adjusting the lid to allow some steam to escape.
Once done simmering, use tongs to remove the bones, skin, cooked vegetables, and herbs from the pot. Discard these after they've cooled.
Strain the broth through a fine mesh strainer into another pot to remove any remaining debris.
Use a ladle to transfer the broth into storage containers. Allow the broth to cool completely, then seal the containers with lids.
Store the broth in the fridge for 3 to 4 days or in freezer-safe containers for several months.