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 eggs with granulated sugar and light brown sugar until smooth and creamy.
Add the unsweetened applesauce, vegetable oil, and vanilla extract to the egg mixture and mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the grated carrots, chopped walnuts, and raisins until evenly distributed throughout the batter.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 35 to 40 minutes, or until a toothpick inserted into the center of the cake 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 and let it cool completely before serving.