For the cake:
3 medium apples
1/2 cup plus 2 tablespoons granulated sugar
9 tablespoons plus 1 teaspoon unsalted butter, at room temperature
1/2 teaspoon vanilla extract
3 large eggs
1 1/2 cups all-purpose flour
2 teaspoons baking powder
1/4 teaspoon fine salt
1 to 2 tablespoons demerara (raw) sugar
For the schlagsahne (whipped cream):
2 cups cold heavy cream
4 teaspoons granulated sugar
1 1/2 teaspoons vanilla extract
Arrange a rack in the middle of the oven and heat to 350°F. Line the bottom of a 9-inch springform pan with parchment paper; set aside.
Peel, core, and quarter the apples. Thinly slice each quarter lengthwise without cutting all the way through to the core side, leaving the quarter hinged together. Finely grate the zest of the lemon into the bowl of a stand mixer fitted with the paddle attachment and set aside; save the lemon half.
Place the sugar and butter in the bowl with the lemon zest and beat until light and fluffy. Stop the machine and scrape down the sides; add the vanilla extract and 1 of the eggs. Beat until combined before adding the second egg. Beat until combined and then add the third egg. Scrape down the sides. Beat until combined.
In a separate bowl, whisk together the flour, baking powder, and salt. With the mixer on low speed, beat in the flour mixture until just combined, and beat in the juice from the reserved lemon half. Scrape down the sides with a spatula and give the batter one last mix by hand.
Scrape the batter into the prepared pan and smooth the top. Gently press the apple quarters into the batter, core-side down, leaving only a bit of space between each quarter. If you have any apple left over, break off slices and insert them into any open space available. Sprinkle the top evenly with the demerara sugar.
Bake, rotating the pan halfway through baking, until the cake is golden-brown and a tester inserted into the cake (not apple) comes out clean, 35 to 40 minutes.
Place the pan on a rack to cool for 5 minutes before running a knife around the edge and removing the springform ring. Let cool to room temperature.
When ready to serve, make the whipped cream: place the cream, sugar, and vanilla in the bowl of a stand mixer fitted with the whisk attachment. (Alternatively, use a large bowl and hand mixer.) Beat on medium-high speed until stiff peaks form. Serve the cake at room temperature with a spoonful of the whipped cream.