Preheat the oven to 350°F (175°C). Grease and flour two 8-inch round cake pans or line with parchment paper.
Grate 3 medium carrots using a food processor grating blade or grate by hand using a hand held grater, set aside for later.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a large bowl, whisk the granulated sugar, brown sugar, applesauce, oil, eggs, and vanilla until well combined.
Gradually add the dry ingredients to the wet ingredients, stirring until just incorporated.
Fold in the grated carrots.
Divide the batter evenly between the prepared cake pans and bake for 30–35 minutes, or until a toothpick inserted in the center comes out clean.
Let cakes cool in the pans for 10 minutes, then transfer to a wire rack to cool completely.