Adding some info on the CO2 method... I have no idea how humane it is for chickens. My experience is only on small mammals after a certain age, as the younger they are, the more resistant they are to this type of euthanasia. With very young rodents, cervical discolation is generally accepted as a much more humane method. I assume the method is the same, perhaps naively, but I don't know if chickens are more resistant, as baby rats might be, due to how much oxygen they need or don't need to function regularly. CO2 is widely accepted as a humane form of euthanasia for many animals (and in Merck Veterinary Manual), but ONLY if done properly. Dry ice can be used, but is very difficult to correctly regulate. I've always used a paint-ball canister filled with Co2 and an adjustable valve to control how much CO2 is pumped into the chamber. The end of exit tubing is placed in a very thin bag, or latex glove, or glass of water, so that you can visually see the amount of air getting pushed out. There are actual meters that will tell you the amount, and that are more accurate, but my methods have worked well for me.
Inhumane: pumping too much CO2 too quickly, causing the animal to be aware that it is suffocating, resulting in panic.
Humane: think about mountain climbers gradually adjusting to less and less oxygen as they climb. The method should be similar. You don't just pump the chamber full of CO2, flushing all O2 out. Rather, you very very gradually increase the CO2 in a chamber that is large enough for the animal to comfortably stand/sit/turn around, and only increase the flow of CO2 after the animal starts to "settle in". Different animals can take longer, but you should never see the animal starting to panic, or gasp for air, or flail in the tub. IDEALLY: you have an animal that settles in to the ground in a natural resting position (I suppose for chickens this may be a bird who has intentionally tucked its head into its feathers), and who closes its eyes (though sometimes they don't) and whose breathing becomes perhaps deeper, but not strained. Once the animal appears to be unconscious, you let the CO2 run for a few minutes longer, before finally increasing the valve to the point where the chamber is 100% CO2. You then leave the chamber at this point for several more moments until there is no possibility that the animal may come back alive.
The last step is the one that i am least sure of as far as CO2's application for chickens. I know how long rats/mice at different ages take before they are guaranteed to be deceased. Knowing how differently birds (and reptiles) can metabolize chemicals or survive oxygen deficiency, I have NO idea when it is 'safe' to remove the bird from the chamber. CO2 for many reptiles, and snakes specifically, can result in a snake that appears dead for 2 days, even to a professional, but later 'wakes up' and recovers showing no adverse symptoms. I would hate for somebody to try the CO2 method as an effort to humanely euthanize/cull/harvest their chicken, only to find out this method is much more difficult than it is for other species, or that it is in fact less humane than other seemingly more brutal methods, due to the possibility for human error.