First make the pastry. Pulse the flour and butter in a food processor until the mixture resembles breadcrumbs or put the flour in a bowl and rub in the cubes of butter with your fingertips. Add the sugar and the cold water and mix or process the mixture into a dough. Wrap the dough in cling film and leave it to chill in the fridge while you prepare the filling.
Peel and core the apple and cut them into thick slices, then put them in a pan with lemon juice and sugar. Bring to a simmer over a low heat and cook for 2 minutes only - don't cook them too long. Tip them into a sieve over a bowl to drain the juice, then set aside to cool. Fold in the cinnamon.
Preheat the oven to 200°C/Fan 180°C/Gas 6. Grease a 22cm shallow pie dish with a little butter. Cut the pastry in two - one piece slightly bigger than the other and put them on a lightly floured work surface. Roll out the smaller piece to fit the pie dish, allowing enough to overhang the sides. Roll out the other half, ready to put over the top of the pie.
Fill the lined pie dish with layers of cooled apple slices, pilling them up high. Be patient about doing this and it might seem ridiculously high but, trust me, this works. Dot with the remaining butter. Brush the pastry edges with some of the beaten egg mixture, place the pastry lid on top and press all round to make sure the edges are well sealed.
Using a sharp knife, trim off the excess pasrty. Brush the pie with the rest of the beaten egg and cut a steam hole, about 5mm in diameter, in the middle. Dust with caster sugar, then bake the pie in the preheated oven for 40-45 minute, until golden.