The flayed man is the sigil of House Bolton, so it isnt just Ramsay who partakes in this particularly gruesome act. However, he does seem to have a certain relish for it. He flays the Ironborn at Winterfell, and more of them at Moat Caitlin despite promising them safe passage upon their surrender. He also flays parts of Theons fingers while torturing him. He appears to delight in this, partly because he enjoys causing others to suffer pain, but also perhaps because he feels it makes him more like a true Bolton, and is something he needs to do in order to prove himself to his father. On the other hand, it is difficult to imagine Joffrey so much as stabbing a person, or doing any sort of real damage himself that involves being in close quarters and getting his hands dirty, much less flaying someone.