Have you asked the advice of Ernie Anastos?
Since the chicken will let you catch and hold it, do so and then put it in a 5 gallon bucket. (You’ll also need a piece of plywood big enough to cover the top of the bucket.)
Ater you put the bird in the bucket, put the plywood on top, slide it over enough to make a crack and then spray in some ether (starting fluid). About 10 seconds or so of spraying will do it. Slide the plywood back and wait about 10-15 minutes, the bird should be deceased at that point.
This is about as ‘humane’ a method, as I know of.
I know that this method works, it’s also very un-traumatic. The animal just gradually loses consciousness and then expires.
“Sir, our scouts report a tasty giant bird caracss on the kitchen table.”
Correcting my typo’s, and linking to the original FFFB strip referred to by the OP and myself.
*https://mycotopia.net/forums/general-discussions/74603-happy-thanksgiving.html
*
Not the OP. The other guy.
“I’m alive
I’m dead.
I’m the farmer,
Killin a chicken.”
Thanks for the earworm, pal…
Cleaning out the wound very thoroughly (use a baster, lavage with very dilute betadine solution) and a course of avian antibiotics might pull her through. Pet stores with a large bird population will sell you some. Put her in a box and keep her warm, clean, and dry. Go on feathersite.com or backyardchicken to get a lot of advice.
Otherwise just catch her and wring her neck. Chicken necks wring very easy. Hold the bird under one arm, grasp her neck just behind the head between the first two fingers of your other hands and jerk hard. Keep hold of her until she stops flapping (a few seconds).
I had a roommate who could have used that story as a script for the “Old Time Thanksgiving” he prepared for us. He even copied the stuffing part.
Drop the bird out of a helicopter hovering over a strip mall parking lot?
Put it in a box and pretend she’s Schrödinger’s Chicken. There’s a 50% chance you won’t have to kill it at all.
Dude, I’m guessin’ Brewster the Rooster.
Haven’t thought of Brewster since I was like 9. Well played, sir. 
“I’m not dead, I’m waiting for them to land!”
The recommended method is by hand, with a sharp knife. Well, it works for turkeys, anyway.
I wasn’t thinking of that till now :mad: 
That doesn’t always work. You might wind up with a zombie chicken.
That’s crazy. It didn’t need stuffing,it wasn’t empty.
Juuuuuuust in case this thread comes up in someone’s google-search:
Neck-wringing:
[ul]
[li]You are trying to accomplish 2 things swiftly: dislocation of the neck bones, and rapid hyperextension of the spinal cord.[/li][li]It is possible to do this with finesse and speed, but be sure to remember the hyperextension part.[/li][li]There will be deep crunching. Do not think about this during the process as it will distract you from the hyperextension part.[/li][li]If you’re new to it and do not fully understand the mechanics of a chicken’s central nervous system, sacrifice a measure of finesse in favor of exaggerated, enlarged whipping/twisting motions. Think: crank-starting an old-timey automobile. The chicken’s body will provide the necessary outward force to hyper extend the spinal cord, and the cranking motion will sperarate the vertibrae.[/li][li]The head might come off in the 0.4 seconds it takes to do the cranking thing.[/li][li]If one of the other hens walks in on you during an execution, the following day you will have one egg from each of the remaining hens, plus one more, all properly placed in the nesting box rather than scattered throught the house and run.[/li][/ul]
After killing the chicken, I was cleaning it. Well, I don’t eat chicken livers, so I threw it to the dog. After that experience, the dog learned he didn’t need me to do the killing, and he helped himself to the chickens. It is very hard to break a dog of this.
I, personally, have done the cranking thing a couple times. It works. Head comes off, body of chicken may run around for a bit. The OP of this thread clarified in post 9 that it was a family pet and “it needs to look like she just didn’t pull through”. A severed head isn’t going to be mistaken for “the wing damage was too extensive”.
In which case, I think post # 16 (plastic bag + tailpipe) might be the best solution.