What kind of Token? ERC20? Or do you mean a coin (which is something with it's own blockchain)? Tokens generally exist on platforms (WAVES, ETH, EXP, UBIQ, NEO, etc...).
If you want to create a token on a platform you don't need to "break it off" anything. You just create a token. You could - presumably - write a smart contract that "forks" a coin and air-drops an equivalent number of all of the "fork" to people holding the original coin.
What I think you're asking about, though, is forking another coin's blockchain. Yes. That's possible. It just takes a lot of work.
Never understood the question in the first place,but am sure what you answered was exactly what he wanted to ask.I have learned some from your explanation.Kudos.