Combine the carrot juice and ginger in a medium saucepan and bring to a boil over medium heat. Boil until reduced to 3/4 cup, about 25 minutes. Let cool to room temperature.

Position a rack in the center of the oven and heat the oven to 325°F. Butter and flour a 8 individual mini loaf pan or 12 muffin tins.

In a medium bowl, whisk the flour, allspice, cinnamon, star anise, baking soda, and salt. In a large bowl with a hand mixer or in a stand mixer fitted with the paddle attachment, cream the butter and brown sugar on medium speed until light and fluffy, 1 to 2 minutes.

On medium-low speed, add the eggs one at a time. On low speed, alternate adding the flour mixture and the carrot reduction in two additions each. Mix each addition until just combined.

Spoon the batter into the prepared moulds. Bake the cakes until a cake tester inserted into the centers comes out clean, 20 to 22 minutes.

Cool the cakes on a rack for 10 minutes and then carefully invert to remove from the molds. Cool the cakes upright on the rack. The cakes may be served warm or at room temperature.

Bring the sugar and 2 cups of water to a boil in a medium saucepan over high heat. Add the carrots, cinnamon stick, and star anise.

Reduce the heat to maintain a simmer; simmer gently, stirring occasionally, until the carrots are translucent, soft, and slightly sticky to the touch, about 30 minutes. Discard the cinnamon and star anise. Remove the carrots with a slotted spoon and set aside. Reserve the syrup to soak the cakes.

To serve, soak the cakes in the syrup for up to 15 minutes. Plate and garnish with the candied carrots and pistachios.