Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, ground nutmeg, and ground ginger until well combined.
In a large bowl, beat the granulated sugar, light brown sugar, unsweetened applesauce, and vegetable oil until smooth.
Add the eggs one at a time to the wet mixture, beating well after each addition. Stir in the vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Fold in the grated carrots. If using, gently fold in the chopped walnuts and crushed pineapple.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 38 to 42 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Turn the cake out onto a wire rack and let it cool completely before frosting or serving.