Preheat the oven to 350°F (175°C).
Grease a 9x13-inch baking pan or line it with parchment paper.
In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a separate bowl, whisk together the brown sugar, granulated sugar, eggs, oil, vanilla, and sour cream until smooth.
Gradually add the dry ingredients to the wet ingredients and mix until combined.
Spread the batter evenly into the prepared pan.
Arrange the apple slices evenly over the top of the batter.
Bake for 35 to 40 minutes or until a toothpick inserted in the center comes out clean.
Allow the cake to cool completely in the pan.
Whisk together the powdered sugar, milk, and vanilla until smooth.
Drizzle the glaze over the cooled cake.
Slice and serve.