Preheat the oven to 350 degrees Fahrenheit (175 degrees Celsius) and grease a 9x9 inch square baking pan.
In a stand mixer, cream together 0.5 cup softened butter and granulated sugar until light and fluffy.
Beat in the egg and vanilla extract until fully incorporated.
In a separate bowl, whisk 1.5 cups flour, baking powder, and salt.
Gradually add the dry ingredients to the butter mixture, alternating with milk, mixing on low speed until just combined.
Spread the batter evenly into the bottom of the prepared baking pan.
Arrange the apple slices in neat rows over the batter, overlapping slightly.
Prepare the crumb topping by combining brown sugar, 1 cup flour, and 1 teaspoon cinnamon; cut in the cold cubed butter using a pastry blender until the mixture forms coarse crumbs.
Distribute the crumb topping evenly over the apple layer.
Bake for 40 to 45 minutes, or until a cake tester inserted into the center comes out clean.
Whisk the powdered sugar, apple cider, and 0.25 teaspoon cinnamon in a small bowl until smooth.
Drizzle the glaze over the cake while it is still warm, then allow to set before slicing.