Preheat oven to 176 °C (350 °F). Grease two 22-23 cm (9'') cake pans with butter or cooking spray (you can also line the bottoms of the pans with baking paper). Set aside.
In a large bowl whisk together flour, baking powder, baking soda, cinnamon, ginger, nutmeg and salt, until well combined.
In another bowl, whisk together oil, eggs, brown sugar, granulated white sugar, applesauce and vanilla extract, until fully combined. Add the grated carrots (drain the grated carrots of the juices, but do not squeeze them) into the sugar mixture and mix until well combined.
Add the wet ingredient mixture into the dry ingredients mixture and mix with a whisk or a rubber spatula until just combined (do not over mix.)
Spread the cake batter between the two pans evenly.
Bake at 176 °C (350 °F) for 30-35 minutes or until the tops of the cakes are set, and a toothpick inserted into the center of the cake comes out clean.
Remove the cakes from the oven and cool in the pans for about 20-25 minutes. After the cakes have cooled a bit, you can remove them from the pans and cool completely on the wire rack.