For your carrot problem, get the really fat bulk carrots, which (at my grocery store anyway) are not woody and very tasty. Peel it as usual, noticing you only have to peel one or two. You’re now perfectly set for your first cucumber solution, cutting it in half lengthwise, or maybe even quarters because carrots are a firmer bite. Personally I start cutting off the fat end and work towards the tip, because you have more to grip for better control on the harder slices, and if your last ones are a little thicker they’re also small.