Heat the oven to 375 degrees. Lightly coat two baking sheets with cooking spray. Use a vegetable peeler to peel away the carrots' outer layer; discard. Peel the rest of each carrot lengthwise into thin strips.
In a large bowl, use your hands to toss the carrot strips with the sugar, salt, and cinnamon until coated.
Evenly divide the strips between the baking sheets, being careful not to overlap. Bake for 15 minutes, switching the position of the pans halfway through. Turn off the oven, open the door slightly, and let the strips remain inside for 10 minutes more.
Meanwhile, stir together the ingredients for the yogurt dip. Serve the crisps with the dip on the side.