When I was a kid I found a 50s book at my grandma's country house that detailed pickling recipes, making preserves, smoking fish and meats and all other things one might need to do at a country house. One of those things was how to slaughter a chicken and I still remember the illustration, hand drawn and very detailed.

Apparently there is an organ at the back of chicken's throat that is saturated with blood, so the technique was to prop the beak open and pierce that "sack" with an awl. The description then explained that this killed the bird instantly and helped containing all the blood inside, keeping things neat and tidy. A cherished childhood memory for me, but perhaps something actionable for you, for the next time? :)