The night before you make the pastries, wrap 120g butter in foil and freeze until solid. Combine the flour, icing sugar and ¼ tsp salt in a large bowl. Add the chopped butter and rub into the flour mixture until it resembles fine breadcrumbs. Stir in 7 tbsp very cold water (if it still seems a little dry, add an extra 2 tbsp). Lightly knead the dough until it just comes together, then roll out on a lightly floured surface into a rectangle. Using a coarse cheese grater, grate half the frozen butter over the middle and bottom thirds of the pastry. Fold down the top third and fold up the bottom third, as if folding a letter. Turn the pastry 90 degrees and repeat with the rest of the frozen butter. Wrap and chill for at least 30 mins.
Put 75g of the pistachios in a heatproof bowl. Cover with boiling water, then leave to soak for 5 mins. Drain, tip into a food processor and blitz for 5-10 mins until you have a smooth paste. Set aside.
Put the milk and double cream in a small pan set over a medium heat. Bring to a simmer, then remove from the heat. Whisk the egg yolks, caster sugar and flours together in a bowl. Gradually whisk in the milk mixture in several additions, then pour into a clean pan set over a low heat. Cook, stirring continuously, until the custard is thick. Simmer for 2 mins more, then stir in a small pinch of sea salt. Transfer to a bowl and leave to cool. Once cooled, tip into a food processor with the pistachio paste and blitz until smooth. Put in a bowl and set aside.
Heat the oven to 200C/180C fan/ gas 6. Put the plums in a roasting tin, cut-side down. Sprinkle over the sugar and vanilla. Roast for 15 mins until softened but holding their shape. Set aside to cool. Turn the oven up to 220C/200C fan/gas 8.
Roll the chilled pastry out on a lightly floured surface to a rectangle of about 38 x 24cm. Trim the sides with a sharp knife, then cut the pastry into eight 11 x 10cm rectangles. Score a 1cm border around each, then put on a baking tray lined with baking parchment (you may need two trays). Brush with the beaten egg, then chill for 15 mins. Bake for 20 mins. Push down the centres, then pipe or spoon in the pistachio custard and top each with a plum half. Bake for 10-15 mins more. Finely chop the rest of the pistachios and scatter them over the warm pastries. Leave to cool completely before serving.