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WARNING: this is graphic. STOP now if you are sensitive.
First, bind the bird so it cant wiggle around and turn it upside down. This is often done with special wing binders, or a killing cone into which the bird will only fit head down. In the cone, it's wings are tightly held against its body, the feet protruding from the top and the head/neck hang out the bottom. A cut-down traffic cone hung on the wall works, or you can buy specially made metal cones.
[[[ My grandmother, a good Norwegian farm matron, taught me how it was done. Normally she just whacked them unceremoniously on a stump with a broad-faced hatchet she had just for the purpose. But on occasion, she did it this way by holding the bird head down on her lap, between her legs. She was good at this and could do it all in a few seconds. I never got good at it.]]]
Next with a sharp, slightly hooked knife about three inches long, cut a deep incision from INSIDE the mouth, just at the back of the throat. This is where the jugular is nearest the joint between head and neck. You can do the same thing from outside on the neck, too, and that is how the Kosher shochet does it. The chicken will jerk at the cut and thrash around if unbound as it bleeds out from the mouth. The blood is collected in a pan or cup. If you just let them hang there, their thrashing would have blood going everywhere.
When the bleeding slows, quickly turn the knife and force it back and down into the brain cavity of the bird. This will kill it instantly. Allow the final blood draining to occur. Proceed with normal processing. In the kosher fashion, the rabbi then packs the carcass in cold salt for an hour or so to remove body heat and allow a short brining period.
What this method does is remove the blood, prevent meat or bone damaged as can happen with decapitation and loosens the feather muscles instantly, so the bird can be dry picked - none of that scalding mess. It does all this in one quick, almost surgical, operation.
The Jews go one step further and pluck them in cold water, as far as I understand it, probably to facilitate the cooling and make things neater all around.
If you've made it this far, let me say I am NOT good at this and barely manage it. I prefer to give away birds rather than do this. Also, it is something you want to do in a special area outside or in a shed, not at the kitchen sink.
Im told the birds sort of just fall asleep when they hang upside down and of course just "drift off" as they bleed out. I wont comment on that here.
But, sooner or later, it befalls the chicken keeper to do his or her duty and dispatch a bird..