Preheat the oven to 350°F and grease a 9-inch round baking pan, 8x8 square pan, or two 6-inch round cake pans.
In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg and salt.
In a medium bowl, whisk the eggs and then whisk in the applesauce, maple syrup, almond milk, coconut oil, and vanilla. Stir in the carrots.
Pour the wet ingredients into the bowl of dry ingredients and stir until just combined.
Pour the batter into the baking dish and bake for 35 to 38 minutes, or until a toothpick inserted comes out clean. Let the cake cool completely before frosting.
Make the frosting: In a large bowl, beat the cream cheese and butter with an electric mixer until smooth. Add the vanilla and powdered sugar and mix until smooth.