Preheat your oven to 350°F (175°C). Grease and flour an 8x8-inch baking pan.
In a skillet, melt the butter over medium heat. Add diced apples, brown sugar, and cinnamon. Cook until the apples are soft and caramelized (about 5 minutes). Set aside to cool.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a large bowl, cream the butter and sugar until light and fluffy. Beat in the eggs one at a time, followed by vanilla extract.
Add the dry ingredients to the wet mixture alternately with the milk, beginning and ending with the dry ingredients. Mix until just combined.
Pour half the batter into the prepared pan. Spread the apple mixture evenly over the batter, then cover with the remaining batter.
Bake for 35-40 minutes, or until a toothpick inserted in the center comes out clean. Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
In a small bowl, whisk together powdered sugar, milk, and vanilla extract until smooth. Drizzle over the cooled cake.