Grease an 8-inch cake round or an 8x8-inch square pan with oil and line with parchment paper.
In a large bowl, stir together the carrots, applesauce, eggs, oil, and vanilla.
In a medium bowl, whisk together the flours, sugar, cinnamon, baking powder and soda, and salt.
Fold the flour mixture into the carrot mixture, stirring to combine. Pour the batter in the prepared pan, and preheat the oven to 350℉. Let the batter rest while the oven heats.
Place the pan in the oven and bake for 30-40 minutes, until a tester inserted in the center comes out clean, the top is firm, and the edges are starting to come away from the edge of the pan. Remove from oven, place on a wire rack, and cool for 20 minutes in the pan. Turn out onto the rack to cool completely.
Make the frosting: In a large bowl using an electric mixer, whip the cream cheese, butter, and honey together until well combined. Taste and add more honey if you want it to be sweeter. Spread the top of the cooled cake with the frosting.