For the Cake:

1. Preheat the oven to 350°F. Butter and flour a 9-inch springform pan.
2. Place the flour, sugar, baking powder, and salt in the bowl of a stand mixer fitted with the paddle attachment. Blend until mixed together.
3. Add the pieces of butter, and mix on low speed until the butter chunks are distributed throughout the flour, with small, pea-sized pieces remaining.
4. Add the vanilla and egg, and blend well until the mixture resembles chunky cornmeal. Evenly spread the dough on the bottom of the prepared springform pan, gently pressing the dough on the bottom and slightly up the sides.
5. Starting from the outside edge of the pan, arrange the apples in a tight, circular pattern.
6. Transfer the pan to the oven, and bake for 25 minutes.

For the Topping:

1. While the cake bakes, make the topping by mixing together the granulated sugar, melted butter, cinnamon, and egg with a whisk or fork.
2. Once the cake has baked for 25 minutes, remove from the oven and spread the topping evenly over the top of the apples.
3. Bake for another 25-30 minutes, until the topping appears dry. Transfer the pan to a wire rack, and cool for 10 minutes.
4. Run a thin knife around the edge of the pan, and remove the sides of the pan.
5. Serve while still warm, or cool completely and serve at room temperature.