To start the short stock, use the bones from a whole duck. Make sure the bones are cut into approximately 2" -inch pieces.
Heat a pot over medium-high heat and add the oil. Add the bones and cook until evenly caramelized and golden all over, about 20 to 30 minutes. Duck is very fatty so as the fat melts, strain the grease from the pan so the bones caramelize properly.
While the bones cook, prepare the mirepoix. Cut the celery, carrot, leek, onion and apple into approximately 2" -inch pieces.
Once the bones have caramelized, remove and set aside. Add the remaining oil and sauté the vegetables for approximately 20 to 30 minutes, until they start to and caramelize. Once done, deglaze with the white wine and simmer until all of the liquid has reduced, about 3 or 4 minutes. Next, add the chicken stock, thyme, demi-glace and the duck bones. Bring this to a quick boil and then turn the heat down to low. Let simmer for about 1 1/2 hours.
Once the stock has reduced and turned a rich, dark color, it is ready to be strained through a fine mesh sieve. Place the stock into a pot and reduce by about half over medium-low heat or until the sauce has a thicker, sauce-like consistency. This should take about 20 to 30 minutes.
Once ready, check the sauce for seasoning and then set it aside while you prepare the squash.