Preheat your oven to 350°F (175°C). Grease and flour a baking pan.
In a large mixing bowl, whisk together the flour, granulated sugar, brown sugar, baking powder, baking soda, cinnamon, nutmeg, and salt.
Add the applesauce, vegetable oil, grated carrots, and vanilla extract to the dry ingredients. Mix until well combined.
Pour the batter into the prepared baking pan and spread it out evenly.
Bake for 40 minutes or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.