In a bowl combine the flour, butter, egg yolk, a splash of milk and sugar. Knead all ingredients to a firm dough. Chill for 30 minutes.
Meanwhile prepare the topping dough: beat egg yolks with sugar until light and foamy. Beat the egg whites until stiff peaks form. Gently fold egg white mixture and flour alternately into the yolk mixture.
Roll short-crust dough on a lightly floured counter to a rectangle the size of the parchment paper lined baking pan. Cover the bottom of the pan with the dough, tap with fingers into place.
Evenly cover the dough with grated apples, sprinkle with brown sugar, lemon zest, ground cinnamon and rum.
Top the apples with the light topping dough and bake the pie in the oven at 200°C for about 45 minutes.
Cool the pie in the baking pan, cut into equal sized square pieces and dust with powdered sugar (optional).