Place the sliced apples in a pan with the sugar and blackberry liqueur and cook gently over a medium heat for 5-6 minutes, or until the apples are softened. Set aside to cool, then mix in the blackberries.
Preheat the oven to 180C/355F/Gas 4.
Divide the apple and blackberry mixture, leaving aside any juices, among four of the puff pastry rectangles, leaving a 2cm/1in border around the edges of the pastry.
Using a lattice cutter, roll over the remaining pieces of pastry, or alternatively use a sharp knife to cut 2cm/1in slits in staggered rows 1cm/½in apart. Gently pull the pastry so that the slits open up and a lattice shape is shown.
Brush the edges of the filled pastry layer with the beaten egg, then carefully lay the lattice-cut layer of pastry on top. Seal the edges by pressing down, then brush the pastry all over with the beaten egg. Sprinkle over some caster sugar and bake the pastries for 20 minutes, or until the pastry is golden-brown and puffy.