Whisk the flour, baking powder, sugar, and salt in a mixing bowl. Rub in the cold butter with your fingertips for about 3 minutes until the mixture looks like coarse crumbs, then stir in the egg yolks and sour cream and press everything into a smooth dough.

Divide the dough into two-thirds and one-third. Flatten the larger piece into a disk, wrap it, and refrigerate. Shape the smaller piece into a block, wrap it, and put it in the freezer. Chill both for 45 minutes while you make the filling.

Peel, core, and coarsely grate the apples. Cook them in a 26 cm / 10-inch deep skillet over medium heat with the sugar, cinnamon, and lemon juice, stirring often, for about 10 minutes until most of the liquid has evaporated and a spoon dragged across the pan leaves a trail. Sprinkle the potato starch over the apples for the last 2 minutes and stir until the juices turn glossy.

Spread the filling in a shallow bowl and cool for 25 minutes. Warm filling will melt the butter in the dough and make the base greasy.

Heat the oven to 175 C / 350 F with a rack in the middle, and line a 23 cm / 9-inch square baking pan with parchment. Press the chilled dough into the pan in an even layer about 5 mm thick, pushing it into the corners, and prick it all over with a fork. Spread the cooled filling on top. Grate the frozen dough on the coarse holes of a box grater over the apples, covering them with loose shreds and without pressing down.

Bake for 50 minutes until the top is deep golden and the filling bubbles at the edges. If the top browns before 40 minutes, cover it loosely with foil.

Cool the cake in the pan on a wire rack for 30 minutes. Lift it out using the parchment, dust with powdered sugar, and cut into 12 squares.