Make the topping: Peel long ribbons from the carrot using a vegetable peeler, then slice the ribbons lengthwise into 1/4-inch-wide strands (you'll need about 48).
Preheat the oven to 250 degrees F. Bring the granulated sugar and 3/4 cup water to a simmer in a saucepan over medium heat. Add the carrot strands and simmer until soft, about 10 minutes. Drain and cool slightly. Arrange the strands in squiggles on a parchment paper-lined baking sheet. Bake until dry, 25 to 30 minutes. Let cool completely.
Make the cupcakes: Raise the oven temperature to 350 degrees F. Line two 24-cup mini-muffin pans with paper liners. Put the flour, almonds, baking powder, baking soda, salt, cinnamon, ginger and allspice in a food processor and pulse until the nuts are finely ground.
Whisk the grated carrots, vegetable oil, granulated sugar, eggs and vanilla in a large bowl until combined. Stir in the flour mixture until just combined. Divide among the mini-muffin cups, filling each three-quarters of the way. Bake until a toothpick comes out clean, 10 to 15 minutes. Transfer to a rack and let cool 5 minutes, then remove the cupcakes from the pans to cool completely.
Meanwhile, make the frosting: Beat the confectioners' sugar, cream cheese and butter with a mixer until fluffy. Beat in the lemon juice and almond extract. Pipe or spread on the cupcakes. Top with the candied carrots.
Photograph by Con Poulos
Recipe courtesy Food Network Magazine