Preheat the oven to 350ºF (180ºC). Line the bottom of an 8-inch square cake pan** with parchment paper. Grease and flour the insides.
In a medium-size bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger and nutmeg.
In a large bowl, whisk together the brown sugar, granulated sugar, eggs, vegetable oil, and vanilla extract until smooth and well combined. Mix in the puréed apples until incorporated. Stir in the flour mixture just until combined.
Transfer the batter to the prepared cake pan. Bake until a toothpick inserted into the center of the cake comes out clean, about 28-33 minutes. Let the cake cool in the pan for 10 minutes, then remove the cake from the pan and transfer it to a cooling rack.