I looked at the website you supplied but I can't see any tricks for putting chickens to sleep.
So here is mine, you'll need a a twelve inch machete, a wooden chopping block, 18 inches of string and a five month old cockerel (or two).... wait, wait, WAIT !!!... you mean putting them to sleep temporarily. That's different then.
Every bird I've ever handled, except Owls*, will go into some sort of torpid state if you hold them firmly, put your hand over their head/eyes and then gently turn them onto their back (preferably in a vertical plane - I mean head over butt going backwards rather than wing over wing). They will just lie there with their little (or sometimes big nasty and clawed) feet sticking up in the air. Keeping your hand over their eyes helps if it's a nasty bird, especially when it comes to 'waking them up'.
I've only tried this on one Owl, but tried it a few times, but it could be that I had a 'sleep trick resistant owl'.
I notice also that when trying to check-out my cart for a subscription that it's not done over HTTPS, which means my credit card number and the special 3 digit number on the back would be sent over the internet without secure encryption. That could well end up with me being put to sleep using the first method above !!!