Make the cheddar pastry. In a large bowl, whisk together the flour, salt, and caster sugar. Rub the cold butter cubes into the flour using your fingertips until the mixture resembles coarse breadcrumbs — you'll still see small flecks of butter, which is exactly what you want. Stir through the grated cheddar until evenly distributed, then add the ice-cold water a tablespoon at a time, mixing with a fork until the dough just comes together. The dough should feel cool, slightly crumbly, and hold together when pinched. Wrap in cling film and chill for at least 30 minutes.

Prepare the apple filling. While the dough chills, toss the sliced apples with the brown sugar, cinnamon, nutmeg, cloves, flour, and lemon juice in a large mixing bowl. The apples should look glossy and evenly coated, and you'll smell the warm spices immediately. Set aside for 15 minutes — this lets the apples release some of their juices, which will thicken into a luscious syrup during baking.

Roll the pastry. On a lightly floured surface, roll out two-thirds of the chilled dough to a circle about 30cm in diameter and 4mm thick. The pastry will feel a little more fragile than standard shortcrust due to the cheese, so work gently and patch any tears with a damp finger. Lift the pastry carefully over your rolling pin and drape it into a 23cm pie dish, pressing it into the base and sides without stretching. The dough should sit snugly against the dish with a slight overhang. Chill the lined dish for 15 minutes while you roll the remaining dough for the lid.

Fill and top the pie. Spoon the apple mixture into the chilled pastry base, mounding it slightly in the centre — the apples will sink as they bake, so a gentle dome is fine. Roll the remaining dough to a 25cm circle and lay it over the filling. Press the edges together firmly, then trim the overhang to about 1cm and crimp with your fingers or a fork. Cut 4–5 small steam vents on top; these should be clean slits that expose the golden fruit beneath. Brush the entire top crust with the beaten egg glaze — you'll see it turn glossy and amber as it bakes.

Bake the pie. Place the pie on a baking tray lined with parchment paper to catch any drips. Bake at 200°C (180°C fan) for 20 minutes, then reduce the oven temperature to 180°C (160°C fan) and bake for a further 35–45 minutes. You'll know it's done when the crust is a deep, burnished gold with darker brown patches around the crimped edges, and the filling bubbles up through the steam vents with a thick, glossy consistency. The apples should feel tender when pierced through a vent with a skewer. If the crust is browning too quickly, tent loosely with foil after the first 30 minutes.

Cool before serving. Let the pie cool on a wire rack for at least 2 hours before slicing. This resting time allows the filling to set so you get clean slices rather than a runny puddle. The pastry will stay crisp and the cheese flavour will mellow slightly, becoming rich and savoury rather than sharp.