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Preheat your oven to 190°C and lightly grease a pie dish.
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In a pan over medium heat, melt the butter and add the onion. Cook until soft and lightly golden.
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Stir in the flour and cook for one minute, stirring constantly.
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Gradually add the chicken broth, stirring to keep the mixture smooth. Add the milk or cream and let the sauce thicken gently.
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Add the cooked chicken, carrots, peas, potatoes, thyme, salt, and black pepper. Stir until everything is evenly combined.
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Transfer the filling into the prepared pie dish and cover with the pie crust. Seal the edges and cut small slits on top to allow steam to escape.
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Brush the crust with the beaten egg.
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Bake for 35 minutes or until the crust is golden and the filling is bubbling.
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Let the pie rest for a few minutes before slicing and serving.