Preheat the oven to 350°F (175°C). Grease and flour a 9-inch (23 cm) springform pan.
In a large bowl, combine 2 cups (250 grams) of the flour, 1/2 cup (100 grams) granulated sugar, baking powder, and salt.
Add the cold butter cubes to the flour mixture. Using a pastry cutter or your fingers, cut the butter into the flour until the mixture resembles coarse crumbs.
In a small bowl, whisk together 1 egg and vanilla extract. Add this to the flour and butter mixture and mix until a dough forms. If the dough is too dry, add 1 tablespoon of cold water.
Divide the dough into two portions: about two-thirds for the bottom crust and one-third for the topping. Wrap the topping portion in plastic wrap and refrigerate.
Press the larger portion of dough evenly into the bottom and slightly up the sides of the prepared springform pan.
In a separate bowl, toss the sliced apples with lemon juice, cinnamon, brown sugar, and the remaining 1/2 cup (60 grams) granulated sugar until well combined.
Spread the apple mixture evenly over the dough in the pan.
Remove the chilled topping dough from the refrigerator and crumble it evenly over the apples to form a streusel topping.
Beat the remaining egg and brush it lightly over the streusel topping to help it brown.
Bake the cake in the preheated oven for 55 to 60 minutes, or until the topping is golden brown and the apples are tender.
Remove the cake from the oven and allow it to 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.