Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan.
In a large bowl, whisk together the flour, granulated sugar, baking powder, salt, cinnamon, and nutmeg for the cake batter.
In a separate medium bowl, whisk together the eggs, milk, vegetable oil, and vanilla extract for the cake batter.
Pour the wet ingredients into the dry ingredients and mix until just combined. Do not overmix.
In a small bowl, toss the diced apples with the granulated sugar, flour, and cinnamon for the apple filling.
Spread half of the cake batter into the prepared baking pan. Sprinkle the apple filling evenly over the batter.
Top with the remaining cake batter, spreading it gently to cover the apples.
Bake for 40-50 minutes, or until a toothpick inserted into the center comes out clean.
While the cake is baking, prepare the glaze. In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Once the cake is out of the oven, let it cool in the pan for about 10 minutes before drizzling the glaze over the top. Let the glaze set before slicing and serving.
Notes
This cake is best served warm or at room temperature. It can be stored in an airtight container for up to 3 days.