Make the Pastry:

1. Combine the Dry Ingredients: In a large mixing bowl, rub the chilled, cubed butter into the flour and icing sugar using your fingertips until the mixture resembles fine breadcrumbs.
2. Add the Egg Yolks and Water: Stir in the egg yolks, then gradually add the cold water, a tablespoon at a time, until the dough comes together. Be careful not to overwork the dough.
3. Chill the Dough: Wrap the pastry in cling film and chill in the fridge for at least 30 minutes.

Prepare the Filling:

4. Prepare the Apples: While the pastry is chilling, peel, core, and slice the apples. Place them in a bowl and toss with the lemon juice, sugar, cinnamon, and flour.

Assemble the Pie:

5. Preheat the Oven: Preheat your oven to 180°C (350°F) or gas mark 4.
6. Roll Out the Pastry: On a floured surface, roll out two-thirds of the chilled pastry to fit your pie dish. Line the dish with the pastry, allowing a little overhang. Prick the base with a fork.
7. Add the Apple Filling: Spoon the apple mixture into the pastry-lined dish, spreading it out evenly. Dot with small pieces of butter if you want a richer filling.
8. Roll Out the Top Crust: Roll out the remaining third of the pastry for the top of the pie. Place it over the apples and crimp the edges to seal. Trim any excess pastry.
9. Brush and Cut Vents: Brush the top of the pie with the beaten egg to give it a golden color. Cut a few small vents in the top to allow steam to escape during baking.

Bake the Pie:

10. Bake the Pie: Place the pie in the preheated oven and bake for 40-45 minutes, or until the pastry is golden and the apples are tender.
11. Cool and Serve: Allow the pie to cool slightly before serving. Enjoy warm with a scoop of vanilla ice cream, a dollop of custard, or a splash of cream.