Take out a large stock pot and place the turkey carcass pieces in it. You may need to break the bones up a bit to get them to lay in the pot without peeking over the lip of the pot.
Peel and chop the onion into quarters with a large sharp knife and cutting board. Peel and chop the shallot in half and place both the onion and shallot in the stock pot. Trim the ends of the carrot off with the knife and cut it into roughly one inch pieces. Cut the celery into three or four pieces and put both the carrot and celery into the pot.
Add the mushroom stems, fresh parsley, fresh thyme, dried bay leaf, and kosher salt into the pot. Use a measuring cup to pour the water in as well, making sure it covers all the ingredients.
Place the pot on the stove and cover it with a lid. Turn the heat to high and bring the ingredients to a boil. Once the water starts boiling, turn the heat to low and allow to simmer, partially covered, for 3.5 to 4 hours.
Once the broth is done cooking, take the pot off the heat and use a pair of tongs and/or a slotted spoon to remove any remaining bones, vegetables, or herbs. I usually do this by setting a large plate or container right next to the pot and placing the remnants in there. You can throw these away.
Next place a fine mesh strainer over a different large pot and pour the broth into that pot, straining any last bits of bones or vegetables out. Take out your storage containers and pour the broth into them using a ladle or a funnel. Let the broth containers cool completely before putting them in the fridge or freezer.