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.