This sure is a hot topic huh, took me awhile to add my 2 cents worth 'cause I didn't want to offend, but here goes anyway.
When I was a kid we fed the cats on top of the deep freeze, mom moved the bowl and got something out and put the bowl back, sure, a kitten had jumped into the freeze unnoticed until I walked by and heard a strange noise in there..I looked, the kitten was almost frozen, posture stiff, frost all over it and worst of all, eyeballs were white clouded and frozen first....it was still alive and mewing pitfully....I screamed for my brother who quickly dispatched the poor thing - with a rifle butt to the head.
Anyone who works in the medical field for any length of time has had the discussion - how do you want to go, and what illness/injury do you want to pull the plug. Trust me, no body wants to linger and suffer in pain. Most all want the big one, and to be unconscious immediately
If you go to the vet, be tough enough to stay with the animal...I worked at a place (for a very short time) that euthanized without anesthia if no one was around- to save money I guess...they didn't just go to sleep...horrid.
If you've ever been knocked out, You know there's no pain, until you wake up! I gotta say for me, if a chick is deformed or the mama hen is trying to kill it, I'll go along with culling -sometimes nature knows better than me...then there are those breeders who cull an animal cause it doesn't met breed standards, I once got a great dog for free because he had too much white on his toes....on condition he would be altered.
On the other hand, a funny chicken takes more resources time and energy to take care of and likely will pass on its deformity, back to natural selection for me...how to dispatch it remains with you
I used to plunge live lobsters into the boiling water...until I heard them scteaming, now I plunge a knife into the brain first. no screaming. I haven't had to kill anything yet (although my rooster has gotten me pretty close to it) I'm happy for the description in good detail on how to exactly wring a neck, lights out no suffering, it may be difficult but I owe it to them, a good life and a good death.