Did Morgoth feed Carcharoth his OWN hand?

I've always found this sentence to be kinda questionable, and personally, I don't believe that Morgoth fed Carcharoth his actual hand, but what do you guys think? I think it'd be cool as hell if he REALLY fed him his hand.

Then Morgoth recalled the doom of Huan, and he chose one from among the whelps of the race of Draugluin; and he fed him with his own hand upon living flesh, and put his power upon him.

Now doesn't it KIND OF sound like Morgoth def Carcharoth his own hand? And upon living flesh COULD mean that upon Morgoth's actual living flesh, and I'm aware Morgoth was a Valar, BUT he could do whatever he wanted with his appearance, and I assume if he wanted to, he could feed the beast his hand if he thought it would impower it.... So what do you guys think?