Pre heat the oven to 350F. Line a 9x9 inch pan with parchment paper, long enough to line the bottom of the pan with extra hanging over the sides. Spray the parchment paper OR grease well with butter.
Shred carrots on the large side of a cheese grater and set aside.
In a large bowl, whisk together melted butter, eggs and sugar.
Then whisk in the greek yogurt and vanilla extract.
Switch to a spoon and stir in the shredded coconut. Then add the flour, baking powder, ground cinnamon, ground nutmeg and salt and stir until combined.
Finally stir in the shredded carrots.
Then, add the batter to the pan and pat down to make sure it's spread out to all the corners.
Bake for 35-40 minutes or until a toothpick comes out clean and the edges are golden and pulling away from the sides of the pan. Leave in the pan to cool. Once cooled, move to a plate and make the frosting. To make the frosting, microwave the cream cheese for 10-20 seconds until warm and quite soft. To the cream cheese add 1/4 cup plus 2 tbsp icing sugar and vanilla extract and stir until smooth.
Frost the cooled cake and then top with additional shredded coconut if desired.
Cake can be stored at room temperature, covered, for up to 3 days.