Grease with oil a small (4-6 cups) bundt cake mold, or line with parchment paper a 9-inch round cake pan.Heat oven to 300 ºF [150 ºC].
Combining dry ingredients
In a large bowl, and with a wire whisk, combine flour, ginger, cinnamon, baking powder, and salt.Set it aside.
Combining wet ingredients
With a stand mixer or handheld electric mixer and with the paddle attachment, beat eggs and brown sugar over medium speed until the mixture is a paler color.Slowly pour the oil into the mixture while still beating.
Making batter
Add the flour mixture and whisk until mixed well. Fold in coconut and carrot using a spatula.Pour the batter into the prepared mold.
Baking
Cook in the preheated oven for 25 minutes or until a toothpick inserted in the middle comes out clean.Remove from the oven and cool to room temperature on a wire rack. Don't turn off the oven and proceed to the next step.
Decorating
While the cake cools down, place the coconut flakes for decoration on a baking tray and bake until most turn golden brown. Remove from the oven and let it cool down to room temperature. Set aside.Combine the powdered sugar with vanilla extract and coconut milk and stir to mix well. It should be very thick but pourable. Add a few more drops of coconut milk if needed.Pour this mix on the cake, and top with the toasted coconut.
Cook's notes
How finely your carrot is grated will change the texture of the cake. Finely-grated carrot will produce a denser, chewier cake, a coarsely-grated carrot will yield a looser cake.