Step 1

Heat the oven to 375°F with a rack in the middle position. Mist a 9-inch springform pan with cooking spray, then dust with flour; tap out the excess. In a small bowl, whisk together the flour and baking powder.

Step 2

In a stand mixer with the paddle attachment, mix the 214 grams (1 cup) sugar, almond paste and salt on low until the paste has broken into crumbly bits, 2 to 3 minutes. Add the butter and mix until combined, about 30 seconds. Increase to medium-high and beat until the mixture is pale and fluffy, about 3 minutes, scraping the bowl as needed. Reduce to medium, then, one at a time, add the eggs, beating for about 20 seconds after each addition.

Step 3

Scrape down the bowl, then add the vanilla and continue mixing on medium until well-combined, about 2 minutes. Reduce to low, add the flour mixture and mix just until the batter is evenly moistened, about 10 seconds; it will be thick. Using the spatula, scrape the bottom and sides of the bowl and give the batter a few folds to ensure no pockets of butter or flour remain. Transfer to the prepared pan and spread in an even layer, smoothing the surface.

Step 4

Slice each apple half lengthwise into ⅛-inch-thick half circles; do not separate the slices. With your hand, gently press down on each half to fan the slices. Divide the fanned apples into 8 equal portions without undoing the fanned effect. Slide a thin spatula or butter knife under the apples, then slide the slices off the spatula near the outer edge of the cake with the slices fanning outward from the center. One at a time, position another 6 sets of slices on the cake, creating an evenly spaced spoke pattern. Place the last set of apple slices in the center. Sprinkle the remaining 2 tablespoons sugar evenly over the top.

Step 5

Bake until the edges of the cake are deep golden brown and a toothpick inserted at the center comes out clean, 50 to 60 minutes. Cool in the pan on a wire rack for 30 minutes. Run a paring knife around the inside of the pan to loosen, then remove the pan sides. Serve warm or at room temperature; dust with powdered sugar just before serving.