I'm considering keeping chickens and have been doing my research on all the factors and have watched several videos of how to slaughter and process the chickens. I'm fully realistic about the necessity of it and feel that if I'm willing to eat the animal, I should have the wear with all to dispatch them.
I came across the gas stun method reading an NYTimes article about some commercial processors beginning to use CO2 for a more humane kill. For a large scale operation, it might not have the benefits of a truly pain free death, but done at home it might work.
CO2 in small concentrations is an analgesic, at somewhat greater an anesthetic and at high concentrations will asphyxiate and kill. The AVMA says that CO2 is the only safe home (ie, not a vet that does house calls, but you doing it yourself) euthanasia method according to the linked article. But you should increase the concentration slowly as immediate exposure to high concentrations will cause burning of the eyes and lungs. You need to put them under but keep their hearts pumping for proper and quick exsanguination for the best taste according to other articles on the internet. Note the CO2 method in commercial settings is criticized because, it's not likely they could process all the birds in time after gassing (which for efficiency they would be gassed by the truckload), invariably some would wake up before throat cutting etc. But if you're at home, you're doing one chicken at a time and can ensure the deed is done before the chicken wakes from its CO2 sleep.
Also, note that you should aim at slitting both the carotids arteries not the jugular. Remember from science class that veins carry blood to the heart and arteries away from. So you want to slit the arteries to stop blood flow to the brain which will render the animal unconscious. And you want to render them unconscious before they wake from the CO2. Cutting the jugular means blood is still getting to the brain and then bleeding out which means the chicken can take minutes to pass out and would be in pain during this time.
Personally, I also believe passage of time is relative. Remember how "time flies when you having fun" but seems to drag on when things are bad. Being in earthquake country, I myself have the subjective experience of earthquakes lasting much longer than the seconds that they actually are. Not to anthropomorphize the chicken, but do we truly know how they experience time and pain?
The linked article provides a crude methodology using vinegar and baking soda. But it gives you the basics of what needs to be done. I would probably invest in a small CO2 tank, the kind used for beer kegs. Even dry ice would be better than the vinegar/baking soda in my opinion. This also allows for exact metering of gas. In brainstorming, I think you could put the chicken in a cone, then the head in a small bucket/cup or 1 qt deli container. The bucket should probably go up to about the shoulders so the head is fully inside the cup. CO2 is heavier than air so you can trickle in the gas into the bucket. The CO2 will fall to the bottom of the bucket mixing with the air and also displacing the air thus increasing in concentration slowly. Once the chicken is under, you would slit it's carotids and let it bleed out using the still beating heart to help in the draining of the blood. I think dry ice could also be very effective and highly controllable and much cheaper than investing in a beer keg tank and the need to go have the tank refilled. Don't know if the cold temp would be stressful to the chicken though.
Anyway, just my preemptive brainstorming of what I might have to do if I decide to keep chickens. I think there would be a market for a chicken retirement home for us urbanites too squeamish to do what needs to be done. After the kill there is still the plucking and evisceration which makes the kill seem like a picnic in my opinion.
Also, I think this anesthesia method might also work for caponing to use all the cockerels that you will hatch but cannot keep. The vet uses halothane to put my parrot under for procedures like microchipping and yet we'll pull out a cockerel's testes with nothing. For me even though they will be exploited for their eggs and eventually eaten, they should at least be treated with as much respect as my parrot.