Preheat the oven to 350°F (175°C). Grease and flour a 9-inch (23 cm) springform pan.
In a large bowl, combine the flour, baking powder, salt, and 1 cup granulated sugar.
Add the cold, cubed butter to the flour mixture. Using your fingers or a pastry cutter, work the butter into the flour until the mixture resembles coarse crumbs.
Add the egg and vanilla extract to the mixture and knead gently until a smooth dough forms. Divide the dough into two portions: about two-thirds for the bottom crust and one-third for the top.
Wrap both portions of dough in plastic wrap and refrigerate for 15 minutes.
In a separate bowl, combine the sliced apples, 1/2 cup granulated sugar, ground cinnamon, and lemon juice. Mix well and set aside.
Remove the larger portion of dough from the refrigerator. Press it evenly into the bottom and slightly up the sides of the prepared springform pan.
Sprinkle the breadcrumbs evenly over the bottom crust to absorb excess moisture from the apples.
Spread the apple mixture evenly over the breadcrumbs in the pan.
Remove the smaller portion of dough from the refrigerator and grate it over the apples using a coarse grater to create a crumbly topping.
Place the cake in the preheated oven and bake for 55 to 60 minutes, or until the top is golden brown and the apples are tender.
Remove the cake from the oven and let it cool completely in the pan on a wire rack.
Once cooled, remove the springform pan sides and dust the top of the cake with powdered sugar before serving.