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 flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger until well combined.
In a large bowl, whisk the granulated sugar, brown sugar, 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, chopped walnuts (if using), and crushed pineapple until evenly distributed.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
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 15 minutes.
Turn the cake out onto a wire rack and let it cool completely before serving.