Preheat your oven to 350°F (175°C). Grease and flour a baking pan.
In a large mixing bowl, whisk together the flour, sugar, baking powder, baking soda, salt, cinnamon, and nutmeg.
Add the vegetable oil, applesauce, and vanilla extract to the dry ingredients and mix until well combined.
Fold in the grated carrots and chopped walnuts (if using).
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.