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 (256 grams) of the flour, 1/2 cup (100 grams) of granulated sugar, baking powder, and salt.
Add the cold butter cubes 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 mix until the dough starts to come together. If the dough is too dry, add 1-2 tablespoons of cold water.
Divide the dough into two parts: about two-thirds for the base and one-third for the topping. Wrap the topping dough in plastic wrap and refrigerate.
Press the larger portion of dough evenly into the bottom and about 1 inch up the sides of the prepared springform pan.
In a medium saucepan, combine the sliced apples, remaining 1/2 cup (100 grams) granulated sugar, ground cinnamon, lemon juice, water, and vanilla extract.
Cook over medium heat, stirring occasionally, until the apples are soft but not mushy, about 10-12 minutes. Remove from heat and let cool slightly.
Pour the apple filling evenly over the dough in the springform pan.
Remove the topping dough from the refrigerator and crumble it evenly over the apple filling.
Bake in the preheated oven for 50-60 minutes, or until the topping is golden brown and the filling is bubbly.
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. Dust the top with powdered sugar if desired before serving.