- Peel layers from carrot lengthwise on one side with peeler until you begin to get wide slices. Peel wide strips, reserving them, until you get about 15 (there will be a few extra). (I just did the whole carrot. I figured that it couldn't hurt to have extras.)
- Bring water and sugar to a boil in a small heavy saucepan, stirring until sugar has dissolved.
- Add carrot strips and simmer, uncovered, 15 minutes.
- Strain through a sieve into a bowl, discarding syrup.
- Let stand 15 minutes.
- Preheat oven to 225 F with rack in middle.
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Line a large baking sheet with nonstick liner (I used parchment paper), then arrange candied carrot slices flat in 1 layer on sheet.
- Bake until dry but still flexible, about 30 minutes. (Leave oven on.)
- Wind carrot strips, 1 at a time, around end of handle of a wooden spoon in a loose spiral, then slip off each curl and return, seam sides down, to lined baking.
- Return curls to oven to dry until crisp, 30 to 45 minutes more. (Some of my curls looked less curly after this step. I'm not sure why it happened - maybe they weren't seam side down, or maybe they got too dry. I loved them anyway.)
- Cool completely on baking sheet. Carrot curls can be made 5 days ahead and cooled completely, then kept in an airtight container at room temperature. Recrisp in a 250 F oven 10 minutes. (Don't put them on your cake until just before serving or they will get soggy.)