This is something I do once in a blue moon with what's left of the carrots used for cooking. I'll put the top part of the carrot in a little bit of water and leave it for a 2-3 days just to see new leaves grow.
Do they eventually grow roots and mature to become new carrots? Well, no... the bottom portion eventually rots. Haven't tried planting it in soil though... but my guess is that it'll rot too.
Perhaps, I can just harvest the leaves...
[My Carrot Crop]