Cake:
Preheat your oven to 350°F (175°C).
In a large bowl, combine the ingredients listed under One: unsweetened applesauce (or oil), granulated sugar, and eggs. Mix until well combined.
Add the ingredients listed under Two: all-purpose flour, baking soda, baking powder, salt, and cinnamon. Mix until just combined.
Stir in the ingredients listed under Three: grated carrots, shredded sweetened coconut, chopped nuts (if using), vanilla, and crushed pineapple (with juice).
Pour the batter into a lightly greased 9×13-inch pan, two 9-inch pans, or three 8-inch pans. For easier cake removal, you can line the pans with parchment paper or use non-stick baking spray with flour.
Bake for 35-40 minutes if using a 9×13-inch or 9-inch pans, and 25-30 minutes for 8-inch pans. The cake is done when a toothpick inserted into the center comes out clean.
Let the cakes cool in the pans for 10 minutes, then remove them to a cooling rack to cool completely.
Cream Cheese Frosting:
In a mixing bowl, beat the softened butter and cream cheese until light and fluffy.
Add the vanilla and gradually beat in the powdered sugar until the frosting is smooth and creamy.
If making a layered cake, invert one cake layer onto a cake plate or stand. Apply a generous dollop of frosting and spread evenly.
Place the second cake layer on top and continue frosting. If using three layers, repeat with the third cake.
Optionally, top with toasted pecans or coconut.
Refrigerate the cake for at least an hour before serving for best results.