Preheat oven to 350°F with the rack in the lower middle position. Prepare a 9-inch springform pan by lightly greasing it with butter, then lining the bottom and sides with parchment circle and strips. This ensures the parchment paper will stay in place, and the cake will easily separate from the side and bottom of the pan with ease.
Cream the butter and sugar over medium speed in a stand mixer bowl fitted with a paddle attachment (or in a medium bowl using a hand mixer) until light and fluffy, about 3 minutes.
125 grams 1/2 cups + 2 1/2 teaspoons) unsalted butter (room temperature), 125 grams (1/2 cup + 2 tablespoons) white granulated sugar
Add vanilla extract and egg yolks. Stop and scrape the sides of the bowl and mix for another 10 seconds.
3 large eggs (yolks and whites separated into 2 bowls), 1 teaspoon vanilla extract
Whisk together flour and baking powder in a separate small bowl. With the mixer on low speed, add the flour mixture in three (3) increments until incorporated, scraping the sides of the bowl at least once. The batter will be very thick, similar to a cookie dough.
200 grams (1 1/2 cups) unbleached all-purpose flour, 8 grams (2 teaspoons) baking powder, 1 teaspoon vanilla extract
Whisk egg whites to stiff peaks and fold a few dollops into the batter to loosen it up, then fold in the rest.
3 large eggs (yolks and whites separated into 2 bowls)
Pour batter into the springform pan and level with a spatula. Carefully place apple quarters core side down into the batter, pressing lightly to secure them in place. You can arrange them according to your desired pattern. Placing one in the center and the surrounding it with the rest is one way to do it.
Bake in the pre-heated oven for 50 to 60 minutes, depending on your oven and pan. It is done when a cake tester inserted in a non-apple section in the centre comes out clean and the top of the cake (not the apples) is lightly browned.
Cool on a wire rack for 5 minutes before unfastening and removing the side of the springform pan. Cool completely on a wire rack before placing on a plate.
1 tablespoon (7 grams) powdered sugar