Preheat oven to 350F and prep your 9x13" pan by greasing with unsalted butter and cooking spray.
Cake Batter Base:
Place your carrots in a food processor and allow it to process them into very fine pieces. You can also grate by hand, but it will take longer.
Separate your egg yolks and whites. Place your egg whites in a clean bowl of a stand mixer or a separate large bowl. Place your yolks into another large bowl. Pour in brown sugar with the yolks and Whip until the color becomes more pale and the mixture becomes thicker. Add in your oil and whip some more. The mixture should ribbon off your whisk into the bowl.
Sift in the cake flour, baking powder, spices and salt all at once. Add in your carrots as well and fold in until just combined. We're going to still add meringue, so some flour being visible is totally fine as we're going to be mixing more and we don't want the mixture to be overmixed.
Meringue and Final Folding:
Begin to whip your egg whites and sugar with your stand mixer or in a large clean bowl and whip until you have stiff peaks to create your meringue.
Working quickly (meringues deflate over time), fold the meringue into the batter in two large additions slowly and gently, folding to make sure the meringue doesn't deflate. You should see no more streaks of white by the time you're done folding.
Baking and Cooling:
Carefully pour the batter into your prepared pan and bake at 350F for 20-25 minutes, checking at the 20 minute mark with a toothpick. You'll know the cake is done when it's risen nicely, is no longer jiggly and a toothpick comes out with only a few moist crumbs.
Out of the oven set the pan aside to cool for 15 minutes.
Milk Mixture:
Combine milks in a large bowl and pour over the warm cake, starting in the center and working your way out. Make sure to pour evenly and don't concentrate on one area.
Let the cake sit in the fridge for at least six hours or ideally overnight to allow the milk to soak into the cake properly.
Icing and Serve:
After the cake has sat in the fridge and when you're ready, make the icing by combining the icing ingredients and whipping the ingredients for five minutes and then mixing with a paddle for another five minutes. Start slowly because if you don't, the powdered sugar will fly everywhere. Increase the speed to medium-high once the sugar is dissolved into the mixture.
Spread the icing evenly over the top of the cake and optionally sift on some cinnamon over the top. Serve and enjoy!