Preheat oven to 350F. Spray one round 8-inch cake pan with non-stick spray.
In a large bowl, add the egg, brown sugar, coconut oil, honey, Greek yogurt, vanilla, cinnamon, and nutmeg, and whisk to combine.
Add the flour, baking powder, baking soda, salt, and fold with spatula or stir gently with a spoon until just combined; don't over mix.
Grate apples and carrots and fold gently to combine into the batter.
Turn batter out into the prepared pan, smoothing the top lightly with a spatula. The batter will be very thick, this is what you want.
Bake for about 35-45 minutes or until the top is golden, the center is set, and a toothpick inserted in the center comes out clean, or with a few moist crumbs, but no batter. Baking times will vary based on moisture content of carrots, apples, climate, and oven variances. Bake until done; watch your cake, not the clock and don't worry if it takes longer to bake than the baking estimates provided.
Allow cake to cool in pan for about 15 minutes before turning out on a wire rack to cool completely before slicing and serving.
To make the frosting, simply mix the yogurt and cinnamon together and top on a completely cooled cake. If you want it a bit sweeter, add a tablespoon or two of honey to the yogurt.