Prepare the Oven and Pan: Preheat the oven to 350°F (175°C). Line the bottom of a springform pan with parchment paper, then grease the bottom and sides with 1 teaspoon of butter.
Cream the Butter and Sugars: Place the room-temperature butter, granulated sugar, and vanilla sugar in a mixing bowl. Using a stand mixer or hand mixer, beat for about 2 minutes, until the mixture is smooth and creamy.
Add the Eggs and Vanilla: Pour in the vanilla extract and add the eggs. Beat again until the ingredients are thoroughly combined and the mixture is uniform.
Incorporate the Lemon Zest: Finely zest the lemon directly into the batter, adding approximately 1 tablespoon of zest. Mix until evenly distributed.
Mix in the Dry Ingredients: Gradually add the flour and baking powder to the bowl. Beat until the ingredients are just combined and a smooth cake batter forms.
Transfer the Batter: Pour the prepared batter into the lined and greased springform pan. Spread it evenly across the bottom so the apple slices can be arranged over the surface.
Arrange the Apples: Peel the apples and cut them into thin slices if you have not already done so. Arrange the slices over the batter, distributing them evenly across the top.
Bake the Cake: Place the pan in the preheated oven and bake for 35–45 minutes, or until a toothpick inserted into the center comes out clean and the cake's edges begin to turn golden brown. The source lists a total cook time of 50 minutes, so begin checking for doneness within the specified baking window. Cool the Cake: Remove the cake from the oven and allow it to cool in the pan for 10 minutes before serving or adding the finishing sugar.
Dust and Serve: Sprinkle the powdered sugar evenly over the cooled cake. For a delicate, even coating, sift the sugar over the top just before serving. Slice and enjoy.