Preheat the oven to 350°F (175°C).
Prepare a 9-inch round cake pan by greasing it with butter and dusting it with flour.
Melt 1/4 cup of butter in the prepared cake pan and sprinkle the brown sugar evenly over the butter.
Arrange apple slices over the brown sugar mixture in the pan.
In a bowl, cream the remaining 1/4 cup of butter with sugar until light and fluffy.
Add eggs one at a time, mixing well after each addition, followed by the vanilla extract.
In another bowl, whisk together flour, baking powder, and salt.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Pour the batter over the arranged apples in the cake pan.
Bake for 35-40 minutes or until a toothpick inserted into the center comes out clean.
Let the cake cool in the pan for 10 minutes before inverting onto a serving plate.
Serve warm or at room temperature, optionally with whipped cream or vanilla ice cream.
