Melt 3 tbsp of butter and let cool. Mix together melted butter, eggs, 3/4 cup of sugar, vanilla sugar and 1 tbsp of lemon juice. Mix together the flour and baking powder and stir into the butter and egg mix.
Cover the bottom of a 10 inch springform pan with wax paper. Melt the remaining butter and let cool slightly, then pour onto the wax-papered bottom. Sprinkle remaining sugar on top.
Peel and core the apples, cut them in half, and then into dice. Cube the Zwieback toasts, and mix together with the apple and the remaining lemon juice, almonds, rum and cinnamon. Add to the spring form pan and spread evenly.
Bake in preheated 350°F oven for about 50 minutes.
Let cool and remove the parchment paper. Heat the apricot preserves until warm, spread over the cake and sprinkle pistachios on top. Serve warm or cold.