Preheat oven to 400°. Salt and pepper the carcass and roast with the carrot, celery, onion and garlic ucovered for one hour.
Transfer roasting pan contents to the slow cooker being careful not to allow fat to go in. Remove the solids first, then you can carefully pour the juices and stop when the fat is about to go in. You want about 1 tablespoon of duck fat in the broth so just spoon it in. Drain fat to a separate container and reserve for another use. Scrape up all the browned bits from the roasting pan and add to the slow cooker. You will still have the dried mushroom and spices in reserve.
Fill the slow cooker with cold water ensuring the carcass is covered. Cook on low for 2 hours. Add the remaining dried mushrooms and spices and cook at least 4 hours and you can easily go 8-12 hours for richer stock.
Turn off heat and allow the broth to cool enough that you can work with it.
Strain out all the solids. Then pass the broth through a strainer lined with a couple of layers of cheese cloth or a coffee filter. Discard the residue collected. Some residue may settle at the bottom of the broth. When using the broth, omit that residue if you want a clear broth.
Adjust salt and pepper to taste and use as you would chicken broth. If you plan to reduce the stock further for a sauce - do not add salt and pepper until you are finishing your sauce.
If you plan to reduce the stock further for a sauce - do not add salt and pepper until you are finishing your sauce. Simmer another 1-2 hours until the concentration is what you want for your sauce.
Broth will keep in the fridge up to a week or freeze up to 9 months. If you pressure can it you can store it up to a year.
Notes
Nutrition is approximate and is for 1 cup of broth.