Preheat your oven to 200°C (fan 180°C). Begin by prepping your vegetables and shredding the cooked chicken. Set all components within reach to keep the cooking process smooth and efficient.

In a large pan over medium heat, melt the butter. Add diced onion, carrot, celery, and potatoes. Cook for 6 to 8 minutes until the vegetables begin to soften, stirring occasionally to prevent browning.

Sprinkle the flour over the vegetables and stir continuously for 1 to 2 minutes to cook off the raw taste. Add garlic, thyme, salt, and pepper. Stir until fragrant, ensuring the flour coats everything evenly.

Gradually pour in the chicken stock, stirring to avoid lumps. Then add milk and stir again. Let the mixture simmer gently for 10 minutes, until the sauce thickens and the potatoes are fork tender.

Fold in the shredded chicken and peas. Simmer for an additional 3 to 5 minutes to ensure everything is well combined and heated through. Remove from heat and taste for seasoning.

Spoon the hot filling into a deep pie dish or four individual ramekins. Let it cool slightly to prevent the pastry from becoming soggy during baking.

Lay the puff pastry over the dish and trim the edges, leaving about 1 cm overhang. Press down lightly to seal. Use a sharp knife to cut a few small slits in the top to allow steam to escape.

Brush the top of the pastry with the beaten egg for a deep golden colour once baked. This step gives the pie a glossy, appealing finish.

Place the pie on the middle oven rack and bake for 25 to 30 minutes until the crust is puffed and golden, and the filling is bubbling at the edges.

Remove from the oven and let the pie rest for 5 minutes. Serve hot with a side of green beans or a crisp salad. A dash of cracked black pepper on top enhances the aroma.