While it's not necessary, try to choose carrots that are somewhat uniform in circumference.
To peel or not-to-peel? It's up to you, if you buy organic carrots. Conventional carrots should be peeled.You're going to cut a series of long, shallow wedges along the surface of the carrot. With your left (or non-dominate hand) hold the carrot on the chopping board. Place your knife against the carrot at a (more or less) 45 degree angle. Press down to make a shallow cut, being careful not to cut all the way through. Turn the carrot around and make a matching cut to meet the first cut. Carefully wiggle your knife to dislodge the carrot wedge and set aside. Repeat five or six times around the carrot, then slice the carrot into coins. As always when using a knife, and especially when cutting down on hard, slippery surfaces: Use common sense and be careful! Curl your fingers under on the hand that's holding the carrot in place. Don't force the wedge -- if it doesn't want to come out, then run your knife through the cuts again to make them a little deeper. You can always recut the wedge shapes once you've cut the carrots into coins, so don't force things. Have fun and experiment. Shallow wedges make a more delicate flower, deeper wedges create a graphic, Marimekko look. Carrot flowers cook-up just like carrot coins, only they're a tiny bit more delicate so use caution when stirring them, especially once they've become cooked and soft.
What to do with the leftover wedges? I usually snack on them while I'm cooking the flowers. Or they could go into stock.
Want to make it even easier? Then drag one of these across the carrot surface instead of cutting wedges. It will produce delicate, frilly-edged carrot flowers.
Related: How To Make Pumpkin Puree
(Images: Dana Velden)