Preheat your oven to 350°F (175°C).
Grease and line an 8x8-inch or 9x9-inch square baking pan with parchment paper.
In a large bowl, whisk together the flour, sugar, baking powder, baking soda, cinnamon, nutmeg, ginger, and salt.
In another bowl, combine the applesauce, vegetable oil, and vanilla extract.
Add the wet ingredients to the dry ingredients, stirring until just combined.
Fold in the grated carrots.
Pour the batter into the prepared pan and spread it evenly.