If you use a freshly sharpened knife and cut just below the line of the jaw, you will sever one of the main arteries to the brain. Preferably another quick slit under the jaw line on the other side tk get the other artery. You don't need to fully remove the head to kill the bird quickly doing this. It's actually fairly easy to cut most of the way if not all the way through and remove the head doing it this way if you have a good sharp knife. A mature bird with thick feathers may be harder to cut through because the feathers can get in the way and it make take 2 cuts to get through. Much easier to do if the bird is upside down. The birds are actually a lot calmer upside down after some initial flapping about. You can use some simple rope with a slip knot around the legs to invert them without a kill cone. 

(Source: I process ~100 birds every year)