Preheat the oven to 350°F and grease a mini cupcake tin.
In a food processor, pulse the carrots until finely shredded. Add the crushed pineapple, lemon juice, vanilla, ginger, and spices. Pulse to combine.
In a large bowl, whisk together the dry ingredients. Pour in the carrot mixture and stir, adding the eggs one at a time. Stir until just combined.
Measure out about 1 - 1 1/2 tablespoons of the batter and scoop into the greased mini cupcake tin.
Bake for 10-15 minutes or until a toothpick inserted in the center of each cupcake comes out clean. For full-size cupcakes, double the cooking time to 25-30 minutes. Cool completely by chilling for at least 2 hours.
While the cupcakes cool, make the frosting by beating together the room temperature cream cheese, lime juice, orange extract, and salt. Once combined, gradually add the powdered sugar one cup at a time, mixing until completely incorporated and the frosting reaches a thick, stiff consistency.
Transfer the frosting to a piping bag or a gallon baggie with the corner snipped off. Leave at room temperature.
When the cupcakes are cooled, pipe a swirl of frosting onto each cupcake. Chill for 1 hour to firm up the frosting or serve right away with softer frosting. Enjoy!