Add the flour, powdered sugar, and baking powder into a food processor (you can also make the pastry by hand) and pulse the dry ingredients for 5-10 seconds to combine them. Then, add in the butter and pulse until the mixture resembles wet sand. Next, add in the egg yolks and pulse until a rough dough forms. Dump the dough onto your work surface and gently knead it until it all comes together to form a smooth dough.
Take ⅓ of the dough and wrap it in plastic wrap and place it in the freezer until needed. With the remainder of the dough, roll it out on parchment paper until it is roughly the size of your cake pan 9x13" (23x33cm) and then flip the parchment paper with the dough over into the pan. Remove the parchment paper from the dough and press the dough to fit neatly and relatively flat on the bottom of the cake pan. You can easily smooth the dough by placing parchment paper over the dough and smoothing it with a pastry roller or the bottom of a measuring cup. Place the cake pan in the fridge while you prepare the filling.
Peel the apples and shred them using a box grater. Place the apples in a colander over a large bowl and squeeze them with your hands to get rid of the excess juice.
Remove the cake pan from the fridge and evenly scatter the shredded apples over the surface of the pastry. Sprinkle the apple with the vanilla sugar, sugar, and cinnamon. Set this aside while you make the meringue.
Add the room temperature egg whites to the bowl of your stand mixer fitted with a whisk attachment. Whisk the egg whites on medium-high speed until they become foamy. Then, begin adding in the sugar a little bit at a time. Whisk until stiff peaks form and they are glossy and shiny and you can no longer feel granules of sugar when you rub the meringue between your fingertips.
Spoon the egg whites onto the shredded apple layer. Using a spatula, evenly spread the meringue and flatten the top.
Using a box grater, shred the remaining frozen pastry evenly over the top of the meringue.
Bake the cake in a 350℉ / 180℃ pre-heated oven for 30-35 minutes. Keep an eye on the cake towards the end, as the pastry tends to brown very quickly. Let the cake cool to room temperature before cutting into it.
