Preheat the oven to 350°F (175°C) and grease two 8-inch round cake pans.
In a bowl, whisk together flour, baking powder, baking soda, cinnamon, ginger, and salt.
In another bowl, beat sugar, eggs, applesauce, oil, and vanilla until smooth.
Gradually add the dry ingredients to the wet ingredients, stirring just until combined.
Fold in the diced apple pieces.
Pour the batter evenly into the prepared pans and bake for 35-40 minutes, or until a toothpick comes out clean.
Let the cakes cool in the pans for 10 minutes, then transfer to a cooling rack to cool completely.
For the frosting, beat butter, powdered sugar, cinnamon, nutmeg, milk, and vanilla until smooth and fluffy.
Frost the cooled cakes generously, and top with a sprinkle of cinnamon.