Put the butter and flour in a food processor with 1/4 tsp salt and blend until the mixture resembles damp breadcrumbs. Or do this by rubbing the butter and flour together in a big bowl with your fingertips. Add the sugar and briefly whizz again or stir to combine.
Whisk the egg yolk with 2 tbsp cold water, and drizzle over the flour mixture. Use the pulse button to blend the mixture once more, keep going until it starts to form larger clumps. If the mixture seems too dry, add a little more water a tsp or 2 at a time, but no more than 3 tsp in total.
Tip out onto a work surface and briefly knead the dough to bring it together into a smooth ball. Avoid overworking or it will become tough. Flatten the dough into a puck shape and wrap well in cling film. Chill for at least 30 mins, or for up to 2 days, or freeze for 2 months.
Next, make the filling. Put the apples and half the sugar in a large bowl, then stir together until the apples are well coated. Set aside for 30 mins to macerate.
Remove the pastry dough from the fridge and divide into 2 pieces, one slightly larger than the other. Re-wrap the smaller piece of dough and set aside. Divide the larger piece of dough into the number of pies you’d like to make, or leave whole for a large one. On a lightly floured surface, roll out the dough to the thickness of a 50p piece, or until large enough to line the base of your pie plate or tin, with a little pastry overhanging. Roll the dough over your rolling pin, lift onto your plate or tins, then press it well into the corners. Scatter the polenta or almonds over the base.
Drain any juice from the apples, then toss through the remaining sugar, cornflour and cinnamon. Layer the apples and blackberries in the lined dish, creating a dome effect in the centre. Heat oven to 190C/170C fan/gas 5 and place a baking sheet on the middle shelf.
Now choose how you’d like to decorate the top of the pie (see tips below). Once covered, whisk the reserved egg white and brush over the pastry. Scatter with extra sugar, then put the pie on the baking sheet. Bake for 25-30 mins for mini pies, 35-40 mins for medium pies, or 45 mins for a large pie, until the pastry is golden and crisp and the juices are bubbling. Cool for 10 mins before serving with cream or ice cream.