Human butchering is pretty annoying nowadays, you can only do it if nearly starving, so feeding the dogs from the corpses of of my enemies is not quite as easy as it used to be. Still,. could be a possibility if- Oh, wait, humane butchering.
Anything small, I go for blunt handaxe to the brain. Midsize, if trapped and close enough to my home, I go grab a grain flail: They are more damaging than they seem, and one to the skull can kill pretty large creatures. Large creatures, piercing to the skull, wait until they bleed to death or the bleeding stops, reapply as necessary. Very occasionally, for very large creatures (bears mostly), cutting the throat might be necessary. The latter might lose in humanity, but definitely win somewhat in efficiency. Otherwise you need something with a pretty large blunt damage, as ending a bear or elk with a strike to the skull is quite hard. Grainflails occasionally work, although the average lack of skill in it doesn't help, but a few of the heavier axes can have a pretty good blunt damage when it comes down to it.