Found in 1.10.9, and I don't know how reproducible this is, but I was able to do it twice in a pen packed full of pigs before I successfully butchered the hare.
Steps I took:
Get several animals in close proximity to each other, like a hare in a pen full of pigs.
Kill one of the animals
Start butchering the dead animal with a knife
While still butchering, point your cursor at a live animal obstructing the dead one
Harvest meat, skin, bones, and fat from the still living animal.