Using a cleaver or kitchen scissors, break up the carcass into several smaller pieces so that they will fit in an even layer in the bottom of a large pot or Dutch oven; set aside.
Heat the oil in the pot over medium-high heat until shimmering. Add the carcass pieces and cook, turning occasionally, until lightly browned all over, about 8 minutes.
Add the remaining ingredients and stir to combine, scraping up any browned bits from the bottom of the pot with a wooden spoon. Bring to a simmer (do not let the stock come to a boil).
Reduce the heat to low and continue to simmer, occasionally skimming any scum off the surface of the stock using a large spoon. Cook, adjusting the heat as necessary to keep the stock at a simmer, until the flavors have developed, about 1 1/2 hours.
Remove and discard any large pieces of carcass. Set a fine-mesh strainer over a 2-quart saucepan and pour the stock through the strainer. Discard the contents of the strainer. (At this point, the stock can be cooled to room temperature and refrigerated in a container with a tightfitting lid for up to 3 days or frozen for up to 1 month.)