The carrot part will not regrow, as that is the root of the plant that you can only grow from seed, but the tops will regrow, and are edible.
I have a bleeding disorder so am always on the lookout for good sources of vitamin K. Carrot greens, which can be bitter because they are also high in potassium, are high in vitamin K. Most people use carrot greens in salads or added to other cooked greens, rather than eating them alone.
But as always, all things in moderation. I am not a doctor and don't want to be sued. Calcium and potassium do a delicate dance within the human body, so if you have medical issues, please check with your doctor prior to using food as medicine.
Often, people think eating things that grow in nature automatically means it is harmless, however there are certain plants that will interact with some medical conditions and/or medications. That is my disclaimer for the rest of the year.