Step 1: Preheat your oven to 350°F (175°C). Grease and flour your 9-inch round cake pans to prevent sticking.
Step 2: In a large mixing bowl, whisk together the eggs, granulated sugar, light brown sugar, canola oil, applesauce, sour cream, and vanilla extract until well combined.
Step 3: In a separate bowl, whisk together the cinnamon, cloves, ginger, nutmeg, flour, baking powder, baking soda, and salt. Gradually add the dry mixture to the wet ingredients, stirring until just combined.
Step 4: Gently fold in the grated carrots until evenly distributed throughout the batter.
Step 5: Divide the batter evenly between the prepared cake pans. Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean. Allow the cakes to cool in the pans for 10 minutes, then transfer to a cooling rack to cool completely.
Step 6: In a large mixing bowl, combine the cream cheese and butter. Beat with an electric mixer until smooth and creamy. Gradually add the sifted confectioners' sugar and vanilla extract. Mix until the frosting is fluffy and spreadable.
Step 7: Once the cakes are completely cooled, place one layer on a serving plate. Spread a generous amount of cream cheese frosting on top. Place the second layer on top and frost the top and sides of the cake with the remaining frosting. Decorate as desired!