I think the previous poster described it adequately... dry ice block, towel, chick in a box. As to the suffering, who is to say except for the chick. I once had to snip a neck for a science experiment and it was disgusting...
You can try putting the chick on dry ice in a box. Cutting the neck with scissors seems to be the method of choice, but I can understand that you are not willing to do that.
Clackclack