No, you can’t regrow a carrot after it gets chopped up.
But you can grow carrot tops! Kiddo is having a bit of fun watching things grow out of a jar, and carrot tops are great cos they grow quite fast. Perhaps there’ll be flowers. And with flowers, come seeds, and that gets you new carrots lol (yah it’s a bit of a stretch…hey, why not?).
“If Life hands you carrots, you can grow a forest!”