Mix together brown sugar, granulated sugar, oil, eggs, applesauce, and vanilla together until smooth and there are no lumps. . In separate large bowl, whisk the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves together until combined. Slowly add the dry ingredients to the wet ingredients and mix together until combined. Gently fold in coconut and carrots. Pour batter into the prepared pan, making sure to divide evenly.