Add the flour, sugar, salt and cold cubed butter to a food processor. Pulse until the mixture resembles coarse meal with small pieces of butter throughout.

Gradually add the cold water while pulsing. Stop when a little dough holds together when squeezed between your fingers. If it remains crumbly, add more cold water, 1 teaspoon at a time, pulsing briefly after each addition.

Turn the dough onto a lightly floured surface, gently gather into a ball and divide in half. Flatten each half with the heel of your hand, fold in half, then fold into quarters and flatten again. Repeat once more, handling gently without prolonged kneading. Shape each portion into a disc approximately 5 inches across. Lightly flour both sides, wrap tightly and refrigerate for at least 1 hour. You will use one disc; reserve the second for another bake.

While the pastry chills, melt 1 tablespoon butter in the cast iron skillet over medium-high heat. Add the shiitakes and cook for 5–7 minutes, stirring occasionally, until lightly browned and their released moisture has evaporated. Transfer to a plate.

Reduce the heat to medium and melt the remaining 3 tablespoon butter in the skillet. Add the onion, carrots and celery. Cook for 8–10 minutes, or until softened, stirring occasionally. Add the garlic and thyme and cook for 1 minute.

Sprinkle the flour over the vegetables and stir for 2 minutes. Gradually pour in the chicken broth, stirring constantly until smooth. Stir in the cream and bring to a gentle simmer. Cook for 3–5 minutes, until thickened.

Fold in the cooked chicken, browned mushrooms, frozen peas, salt and pepper. The filling should be thick but spoonable. Add a small splash of broth if needed, then taste and adjust the seasoning.

Remove the skillet from the heat and let the filling cool for 20–30 minutes, until warm rather than piping hot. Meanwhile, preheat the oven to 400°F (200°C). Keep the pastry refrigerated until the filling is ready.

On a lightly floured surface, roll one chilled disc into a circle slightly larger than the skillet, approximately ⅛ inch (3 mm) thick. Lay it over the filling, trim any excessive overhang, then fold and crimp the edge just inside the skillet rim. Cut 4 small slits in the centre to release steam.

Beat the egg with 1 teaspoon water and brush a thin layer over the pastry. Sprinkle with a little flaky salt, if using. Place the skillet on a rimmed baking sheet and bake for 35–45 minutes, until the pastry is deeply golden and the filling bubbles through the vents. Loosely cover the pastry edges with foil if they brown too quickly.

Let the pot pie rest for 10–15 minutes before serving.