Prepare the springform pan by greasing it with butter or non-stick baking spray. *If your pan isn't non-stick, also line the bottom with parchment paper.
To a medium mixing bowl (or the bowl of a stand mixer), add butter, 1 1/2 cups sugar, eggs, and vanilla extract. Using the stand mixer or an electric hand mixer, beat ingredients together on medium-high power until creamy. Stop mixer and use a silicone spatula to scrape down the sides of the bowl.To the bowl, add flour, baking powder, and salt, and mix on low speed, just until combined.
Spread batter into prepared springform pan, then use an offset spatula or back of a large spoon to smooth the top. Turn on oven to 325° F to preheat while you prepare the apples.
Peel and core the apples. Using a mandolin or paring knife, cut apples into thin slices, about 1/8-inch thick. Either group the apples together (about 1/3 of an apple) and gently press into the cake batter. This results in a slightly firmer apples. Or, for a softer consistency, fan the slices around the cake in a single layer.Sprinkle 1/3 cup of sugar and the cinnamon in an even layer over the kuchen.
Bake cake in preheated oven for 50 minutes, until it the top is golden in color. Serve warm. If desired, lightly dust with confectioners sugar before serving.
Notes
Store in an airtight container at room temperature for up to 2 days, in the refrigerator for up to 5 days, or in the freezer for 3 months.