Preheat oven to 350 °F / 180 °C. Grease the springform pan.
For the dough stir flour, baking powder and vanilla sugar. Cut in butter until crumbly then add egg and milk. Knead quickly to a smooth but soft shortcrust.
Press about two-thirds of the dough over the base of the pan and 1 in up the side. Dock with a fork. Blind-bake 10 min until pale gold. Let cool a few minutes.
Peel core and slice apples. In a saucepan heat apples with sugar, cinnamon, water, raisins and rum just until the fruit starts to soften, 3–4 min. Cool briefly.
Spoon filling onto the pre-baked base. Roll or pat the remaining dough into a circle and lay it over the apples, sealing at the edge.
Brush top with egg wash. Bake 20–25 min until nicely golden.
Cool in the pan on a rack. Serve slightly warm or at room temperature, with whip cream, plain or a scoop of vanilla ice cream.
The crust stays tender the next day and the flavors deepen, making this a great make-ahead cake.