Pre-heat the oven to 200C/180C Fan/Gas 6. Line an 8inch square pan.
In a large bowl, or the bowl of a stand mixer, combine both sugars, milk, apple sauce and vegetable oil Mix together on a medium speed until all combined.
Add in the flour, spices, baking powder and baking soda and mix again until everything is incorporated. Fold in the grated carrots so that everything has come together.
Pour the batter into the baking pan and bake for 35-40 minutes, until a skewer comes out clean. Leave to cool on a wire rack.
Once cool, make the buttercream: Use a high speed mixer to beat the butter and maple syrup together. Then, add half the icing sugar before gradually incorporating the rest until you reach the desired consistency (thick but spreadable).
Spread this over the cake, dust with icing sugar, a little extra syrup, and serve!