Preheat the oven to 190 C. For the cake: Grease a 13x18 sheet pan with butter, line bottom of the pan with parchment paper and generously butter and flour the paper.
In a small bowl, sift together the flour, cinnamon, baking powder, baking soda, ginger, nutmeg, and salt and stir to combine. Place the eggs and granulated sugar in the bowl of an electric mixer fitted with the paddle attachment and beat on medium-high speed until light, thick and pale, about 5 minutes. With the mixer on low, add the carrot, then slowly add the flour mixture, mixing just until incorporated. Pour the batter into the prepared pan and spread evenly. Bake until the top springs back when gently touched, about 15 minutes.
While the cake is baking, lay a clean, thin cotton dish towel out on a flat surface and sift 1/4 cup confectioners' sugar evenly over it. (This will prevent the cake from sticking to the towel.) As soon as you remove the cake from the oven, loosen it around the edges and invert it squarely onto the prepared towel. Peel away the parchment paper. Starting at one of the short ends, gently roll the warm cake and the towel together. Allow the rolled cake to cool completely on a wire rack.
For the candied carrots: Combine the grated carrot, sugar and 1/2 cup water in a small saucepan. Bring to a boil over medium-high heat, stirring often, and cook until the carrots are bright orange, glossy and almost dry, about 12 minutes. Remove to a plate with a slotted spoon and let cool. (The candied carrots will be in clumps.)
For the frosting: Beat the cream cheese, butter and salt together with an electric mixer on medium speed until well combined and smooth, about 3 minutes. Reduce the speed to low and add the confectioners¿ sugar, vanilla and 1 teaspoon milk. Continue to beat until completely combined, about 2 minutes. (If needed add the second teaspoon of milk, the mixture should be fluffy not runny.) Increase the speed to high and beat until fluffy, about 4 minutes more.
To assemble: Place 1/2 cup of the frosting in a small bowl and set aside. Carefully unroll the cake onto a cutting board with the towel underneath. Using a small offset spatula, spread half the frosting evenly over the cake. Sprinkle the frosting evenly with walnuts, raisins and golden raisins, making sure to go all the way to the edges. Disperse teaspoons of candied carrots across the frosting. Carefully re-roll the cake, using the towel as a guide. Place the cake roll on a platter. Frost the sides and ends of the cake with the remaining frosting and press shredded coconut all over the surface.
Combine the reserved 1/2 cup frosting with 2 to 4 drops orange food coloring and mix well. Transfer the tinted frosting to a pastry bag fitted with a small plain tip. Pipe 12 carrots evenly down the center of the cake roll and top each with a small piece of reserved carrot frond. Place the cake in the refrigerator until fully chilled, about 2 hours.