The original expression of chopping the dog's tail by pieces comes from Russia and actually mocks someone not having courage to make a big step, but preferring baby steps, often involving pain to other parties involved.
We could say this of someone who has decided to break up with a partner, but instead of saying it directly, they prefer to be more distant, colder, etc, so that the break up happens "naturally" thinking it is more humane. In fact childishly unaware of the pain they is causing. 

Apparently, the expression disapproves of this cowardly and immature behaviour.