1.
Preheat the oven to 400 degrees. Grease an 8-inch round cake pan all over (including the bottom and the sides). Set aside.
2.
In the bowl of a stand mixer, cream together the butter and sugars (white and brown) until smooth. Add the egg and beat until well combined. Be sure to scrape down the sides of the bowl, as necessary. Scrape in the seeds of the vanilla bean (or add the vanilla extract) and mix briefly to distribute. Don't remove the bowl from the stand mixer yet.
3.
In a medium bowl, sift together the flour, baking soda, cinnamon, nutmeg, and salt.
4.
With the stand mixer running on its lowest setting, add the flour mixture one spoonful at a time until it is fully incorporated into the wet mixture. Add the applesauce and mix to combine.
5.
Pour the batter into the prepared cake pan and spread it out so that it's evenly distributed.
6.
Bake for 25 - 30 minutes, or until a toothpick inserted in the center comes out clean.
7.
Let cool before serving flipping the pan to remove the cake. Sprinkle with confectioner's sugar, if desired.