Lightly toast the walnuts (Spread the nuts in a single layer on a baking sheet lined with parchment paper and toast in a preheated 350°F (180°C) oven until the skins start to darken and wrinkle. When the nuts are cool enough to handle, wrap them in a kitchen towel and rub vigorously to remove skins. Not every speck will come off.) and then coarsely chop; set aside. Peel the carrots and cut into 1/2-inch (12-mm) slices.
Bring a large saucepan three-fourths full of water to a boil. Add the carrots and cook until tender, 10-15 minutes. Drain and let cool. In a food processor, purée the cooked carrots. You should have about 1 cup (8 fl oz./225 ml) purée.
Preheat the oven to 350°F (180°C). Line the bottom of an 8-inch (20-cm) square baking pan with parchment (baking) paper.
Sift the flour, baking powder, baking soda, salt, cinnamon, and nutmeg together onto a sheet of waxed paper; set aside.
In a large bowl whisk together the eggs and brown sugar until well combined. Whisk in the milk and melted butter. Whisk in the dry ingredients and then the carrot purée. Using a large rubber spatula, stir in the walnuts and currants.
Pour the batter into the prepared pan and smooth the top. Bake until lightly browned and a skewer inserted into the center comes out clean, 45-50 minutes. Let cool completely on a wire rack. Run a table knife around the edge of the pan and invert the cake onto a serving plate. Peel off the parchment paper.
To make the frosting, using a stand mixer, beat the cream cheese, butter, confectioners' sugar, and vanilla with the paddle on medium speed until combined. Using a straight frosting spatula, spread the frosting on the top and sides of the cake. Refrigerate the cake if not serving immediately. Remove it from the refrigerator 30 minutes before serving.