Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch 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, and vanilla extract to the dry ingredients and mix until just combined.
Fold in the grated carrots and walnuts (if using).
Pour the batter into the prepared baking pan and spread it out evenly.
Bake in the preheated oven for 35-40 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.