A sharp axe to cut clean through would take one blow.
Some folks use very sharp loppers like you cut shrubbery/tree limbs with. I've never used those for this purpose either. But I can see it being quick and effective if they are properly sharp.
:hugsI'm sorry you face this. It's not easy...
For me, I would not try the broomstick method on a full grown Turkey. Even large roosters I don't use the broomstick method.
For a large bird, I'd decapitate, cut the jugular or shoot the bird. Whatever method is quickest.
Sorry, I'm not familiar with the ether and it seems like it would...