Is it really that inconvenient to clean a carrot? Just peel it and eat it. I can peel a carrot quicker than most people can even decide what to have at a vending machine.
When I want a carrot I grab one out of the fridge, grab the peeler off of the counter, pull my compost box out from under the sink and peel it directly into the compost. Why would you need a cutting board?!