Step 1: Peel
Peel the carrots using a Y-peeler and making a single stroke all the way from the head to the tip to remove the peel in as few pieces as possible. This will help keep the shape of the carrot more uniform.
Step 2: Trim
Trim the blunt end of the carrot off, as well as the tip, if it is still dirty.
Step 3: Cut in Half
Cut the carrot in half crosswise in order to make the pieces a little less cumbersome to work with. Especially large carrots (which for some reason seem to localize in Chinese grocers) may need to be split into three or even four pieces.
Step 4: Quarter
Quarter the largest parts of the carrot lengthwise, and halve the smaller parts. The goal is to get an even thickness for all the sticks. Especially large carrots may need to be cut into 6 or 8 pieces radially.
Step 5: Dice
Cut the carrot sticks crosswise into even small, medium, or large dice depending on how you plan to use it.
Step 6: Faux Tourné
A traditional tourné cut creates a little football with seven sides. It's a bitch to pull off, and entirely unnecessary. Here's how to cut carrots into attractive pieces without bothering. Start by cutting the top of the carrot off at a 45 degree angle.
Step 7: Rotate and Cut
Rotate the carrot a quarter turn and cut again at a 45 degree angle. Continue cutting the carrot this way all the way down its length.
Step 8: Finished Faux Tourné
The finished pieces should be relatively even in size, and much more attractive on the plate than disks or chunks.
Step 9: Julienne and Brunoise
To finely julienne a carrot for more elegant preparations, start by trimming off about an 1/8th inch from one side in order to create a stable base for the carrot to sit on. Use the scraps for stocks, compost, or eat immediately.
Step 10: Cut Planks
Rest the carrot on its trimmed side, and carefully slice it into thin planks with a sharp chef's knife.
Step 11: Julienne
Stack two or three of the planks together and slice into even julienne.
Step 12: Brunoise
To cut brunoise, take a bunch of your julienne and cut crosswise into even, tiny cubes. Getting these perfect is one of the most satisfying things you can do with a knife in the kitchen.
Step 13: All Finished
Medium dice, faux tourné, julienne, and brunoise, ready to cook.