Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
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 until just combined.
Fold in the grated carrots and chopped walnuts until evenly distributed.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 40 to 45 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.
Transfer the cake to a wire rack to cool completely before serving.