Peel and core the apples and chop into small dice. Place in a saucepan with the lemon juice, sugars, cardamom (start with ¼ tsp and add more later if needed) and cinnamon. Bring up to a simmer and cook for about 15 minutes, adding a splash of water if needed, until the apples are tender. Taste, and add more cardamom and sugar as needed. Set aside until completely cold then sift over the flour and stir through.
Place the flour, icing sugar, salt and butter in a food processor and pulse until the mixture resembles fine breadcrumbs. Add the egg yolks and ice water and pulse again until it comes together into a ball. Shape the pastry into a disc, wrap in clingfilm and refrigerate for about an hour.
Line a muffin tray with strips of greaseproof paper so that they overhang either side of each cavity so that you can lift the pies out easily (or use disposable foil tins). Roll the pastry out thinly on a lightly floured surface. Use a 10-12cm (depending on the size of your muffin tin) round pastry cutter to cut out 10-12 rounds (I didn't have a large enough cutter so cut round a bowl with a knife). Press the rounds into the muffin tin, the edge should come up to just above the top of the tin. Fill each one right to the top with the cooled apple mixture then re-roll the pastry scraps and cut into thin strips.
Lay the strips in a criss-cross pattern across the tops of the pies, brush the ends with a little of the reserved egg white and gently press them into the pastry base to seal. Place the tray in the freezer for 15 minutes while you preheat the oven to 200C/400F/gas mark 6.
Brush the tops of the pies with egg white, being careful not to get any between the pastry and the tin or the pies will be difficult to remove. Bake for 30 minutes until golden brown and cooked through; if the tops start to darken too much, loosely cover with tin foil.
Leave the pies to cool completely in the tin, the pastry is delicate when warm and they will be difficult to remove. Once cold, carefully prise the pies out of the tin and serve. Store in an airtight container.