To make the pastry: Put the flour, sugar, and salt into a food processor and pulse a few times to mix it together. Cut the butter into cubes, and add it to the flour. Process the mixture until it resembles fine breadcrumbs, taking care not to over mix it, then add the egg and switch the motor on again until the pastry starts to come together. Once it starts to form one large ball in the machine, tip it out onto a floured surface and bring it together quickly with your hands. (If the pastry is still dry, add 1 tablespoon water and pulse a few times to bring it together.)
Roll the pastry out to around 1/4" thick. Use it to line your pie tin, pressing it firmly into the flutes, then trim the edges, prick the base with a fork, and refrigerate for 1 hour.
Preheat the oven to 400F. After an hour, line the pie dish with foil and fill with baking beans or dry rice or lentils. Blind bake the pie case for 15 minutes, then remove the beans and foil, brush with egg wash, and bake for another 10-15 minutes, until cooked through and golden brown.