Set the oven to 350°F (175°C). Grease two 9-inch round cake pans, then line the bottoms with parchment paper. Lightly grease the parchment so the baked layers release cleanly.

Whisk the flour, baking soda, baking powder, cinnamon, ginger, and salt in a medium bowl. Break up any small lumps so the leavening and spices spread evenly through the flour.

In a large bowl, whisk the eggs, sugar, vegetable oil, vanilla, and undrained crushed pineapple until blended. The pineapple liquid stays in the batter and contributes moisture.

Add the dry mixture to the wet ingredients in portions, stirring gently after each addition. Fold in the shredded carrots, coconut, and chopped nuts with a spatula. Stop mixing as soon as no dry flour remains.

Divide the batter evenly between the prepared pans and smooth the tops. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out with only a few moist crumbs. Let the cakes rest in their pans for about 10 minutes.

Turn the cakes onto a cooling rack and remove the parchment. Leave them until completely cool before adding the frosting. Warm cake can soften the cream cheese mixture and make the layers difficult to finish.

Beat the softened butter and cream cheese until creamy. Add the confectioners’ sugar gradually, followed by vanilla and the pinch of salt. Continue mixing until smooth and thick enough to spread. If needed, let the frosting stand briefly at room temperature until it firms slightly.

Place one cake layer on a serving plate and spread a layer of frosting over the top. Set the second layer above it and cover the top and sides with the remaining frosting. Press the unsweetened shredded coconut gently over the surface. Refrigerate the finished cake before slicing if a firmer frosting is preferred.