Beat the butter until smooth. Mix in the sugar and vanilla sugar, add one egg after the other and stir well. Mix the flour and baking powder and stir in briefly. In a well-greased springform pan with a little batter, form an edge that is almost 4 cm high and distribute the remaining batter on the base.
Heat the applesauce for the filling. Meanwhile, stir the custard powder with a little liquid so that it does not form lumps, and peel, core and cut the apples into small pieces. When the applesauce is hot, add the custard powder and the apple pieces. After stirring for about a minute, pour the hot pudding onto the dough.
For the crumble mix all ingredients with your hands. Then distribute the crumble evenly on the pudding.
Bake the cake in the hot oven at 160 ° C top / bottom heat on the lower rack for about 60 minutes. When it has cooled down a bit, put the cake in the fridge so that the filling sets.