Preheat the oven to 200°C (400°F).
Prepare the filling: Heat the oil in a big skillet over medium heat. Add the chopped onion and sauté until translucent. Add the butter and mushrooms, cooking until the mushrooms are soft.
Thicken the sauce: Sprinkle the flour over the mushroom mixture and stir until the vegetables are coated. Gradually pour in the chicken stock and milk, stirring constantly, until the mixture thickens and bubbles. Season with salt, pepper, and thyme.
Combine the chicken: Add the diced chicken to the skillet, mix well, and cook for an additional 2-3 minutes. Remove from heat.
Assemble the pie: Pour the filling into a deep pie dish. Cover with the puff pastry sheet, trimming any excess. Seal the edges and brush the top with the beaten egg for a golden finish.
Bake: Place the pie in the oven and bake for about 30 minutes, or until the pastry is puffed and golden brown.
Rest and serve: Let the pie stand for a few minutes before serving to allow the filling to set slightly.
