Use a mixer to beat the shredded carrots, eggs, vanilla extract, olive oil, applesauce, and brown sugar until well combined
In a separate bowl, whisk together oat flour, coconut flour, baking soda, baking powder, salt, cinnamon, and nutmeg
Add the dry ingredients to the wet, beating until well incorporated
Grease a 13 by 9 glass baking dish
Pour the cake batter into the dish and smooth it out into an even layer
Bake the carrot cake at 350 degrees for 20-25 minutes, or until a toothpick inserted in the cake comes out clean
Allow the cake to cool completely before frosting
To make the frosting, first start by beating the softened cream cheese until fluffy
Add the powdered sugar and beat to combine
Add the greek yogurt and beat until well mixed. Taste and add more powdered sugar, if desired. I prefer this cake mildly sweet, but you can certainly add additional powdered sugar to your liking for a sweeter frosting.
Frost the cooled cake
Chill the carrot cake covered in the fridge for the best texture
Serve with vanilla ice cream
Notes
If you cannot find oat flour, use a blender to grind up gluten free rolled oats into flour.