1. Preheat oven to 375ºF. Line 2 large baking sheets with parchment. Set a wire rack over a rimmed baking sheet. Sift flour, baking powder and salt onto a sheet of parchment.
2. Using an electric mixer on medium-high speed, beat butter, sugar and zest until light, about 3 minutes. Reduce speed; beat in egg and carrot puree. Scrape down sides of bowl. (Mixture will look curdled.) Stir in flour mixture and currants.
3. Drop dough by tablespoons 2 inches apart on baking sheets. Bake until lightly browned, about 15 minutes, switching sheets top to bottom and front to back halfway through. Let cool on sheets on wire racks for 2 minutes, then remove to racks to cool completely.
4. Make icing: Mix juice and butter. Gradually add confectioners' sugar, beating until smooth. Transfer to a ziplock bag. Snip off a corner; drizzle icing over cooled cookies. Let stand until icing hardens, about 15 minutes.