Coat the duck carcasses with the olive oil and salt well. Set aside for 15 minutes while you heat the oven to 400 degrees. Alternatively, grill the duck carcasses over high heat until well browned. If you are roasting, put the duck or goose carcasses in a large roasting pan and cook at 400 degrees for 30-45 minutes, or until well browned.

Meanwhile, chop the duck feet with a cleaver or heavy chef's knife, or score the pig's foot all over. This opens the feet up to the broth, so all that collagen can get into it. This is what makes the concentrated broth set up solid. Do not skip the feet!

Once the carcasses are roasted, remove and chop them into large pieces with poultry shears or a large chef's knife or cleaver. You're doing this to be able to fit them all into your stockpot. Stuff the duck carcasses into your largest stockpot and cover by 1-2 inches with cold water. Add the feet to the pot.

Bring the stock to a bare simmer - do not let it boil - and skim any froth that surfaces. Do this as many times as needed. Simmer the stock for 4-8 hours. You can stop now if you'd like and pick up the process the next day.

Once the stock has simmered for 4-6 hours, add all the vegetables, herbs and spices. Simmer this for 90 minutes to 2 hours -- no longer.

Strain the stock. First pull all the bones and things out with tongs, then use a Chinese spider strainer or slotted spoon to fish out the smaller bits. Finally, pour the stock through a fine-mesh sieve lined with cheesecloth (or a paper towel) into another large pot. You might need more than one pot. This is another place where you can stop and pick up the process later.

Once you have your finished stock, you should have something on the order of 2 gallons. Now a normal demi-glace recipe would have you split the stock and make a sauce espagnol out of one half, then combine it with the other half after you've already added some more herbs and aromatic things. I see no reason to do this. Instead, move the pot off-center to a burner set on low heat: This will concentrate the impurities you will need to skim off all on one side of the pot. Skim frequently.