Preheat the oven to 350°F (175°C) and grease a 9-inch round cake pan.

In a mixing bowl, whisk together the flour, baking soda, cinnamon, nutmeg, and salt.

In another bowl, mix the applesauce, sugar, vegetable oil, egg, and vanilla extract until well combined.

Gradually add the dry ingredients to the wet ingredients, stirring until just combined.

Pour the batter into the prepared cake pan and bake for 30 minutes or until a toothpick inserted in the center comes out clean.

Allow the cake to cool in the pan for 10 minutes before transferring to a wire rack to cool completely.