Line a baking tray with baking parchment. Dice 250g of the butter and tip into a mixing bowl. Add the flour, 1 tsp salt and 1½ tsp black pepper and rub together with your fingers until the mixture resembles fine breadcrumbs, or use a food processor. Stir in 8 tbsp cold water, then bring the mixture together with your hands to create a firm, smooth dough. Cover and chill for 20 mins.
Cut the dough into six equal-sized pieces and roll each portion into a ball. Chill, covered, for a further 15 mins. Meanwhile, tip the potatoes, apples, cheese and some seasoning into a bowl and toss together.
Lightly dust a work surface with flour, then roll out one of the dough balls to form a 22cm round. Spoon one-sixth of the potato mix down the centre of the round, then pinch the edges together to make a sealed parcel. Repeat with the remaining dough balls and filling. Transfer the pasties to the lined baking tray and chill for 30 mins.
Heat the oven to 220C/200Cfan/gas7. Melt the remaining butter in a small saucepan over a low heat, then generously brush over the pasties. Bake for 20 mins, then turn the heat down to 200C/180C fan/gas 6 and cook for 25-30 mins more, or until golden brown.