Preheat the oven to 400°F / 204°C.

Season the chicken cubes with salt and pepper. Melt 4 tbsp butter in a 12-inch cast-iron skillet over medium-high heat. Add the chicken and cook for 6 to 8 minutes, turning occasionally, until browned outside and cooked through to 165°F/74°C at the center. Transfer to a plate.

In the same skillet, add the onion, carrots and celery. Cook for 5 to 6 minutes, stirring, until the vegetables soften.

Add the garlic and cook 30 seconds until fragrant.

Sprinkle the roux flour over the vegetables and stir constantly for 1 minute to cook out the raw flour taste.

Slowly whisk in the chicken broth, scraping up any browned bits from the bottom of the skillet. Simmer for 5 minutes, stirring often, until the liquid thickens enough to coat the back of a spoon.

Remove the skillet from the heat and let the filling cool for 2 minutes. Stir in the Greek yogurt, thyme, cooked chicken and peas until combined. Adding the yogurt while the filling is still boiling will make it curdle, so wait for it to calm down first.

In a mixing bowl, whisk together the crust flour and salt. Cut in the cold butter with a pastry cutter or your fingertips until the mixture looks like coarse crumbs.

Add ice water 1 tbsp at a time, mixing gently, until the dough just holds together when pressed.

Roll the dough on a floured surface into a round slightly larger than the skillet. Drape it over the filling, tuck the edges in against the inside of the pan, and cut 3 to 4 vents in the top.

Brush the crust with the beaten egg.

Bake for 28 to 32 minutes, until the crust is golden brown and the filling is bubbling around the edges.

Let the pie rest for 10 minutes before cutting into wedges to serve.