Heat the oil in a large frying pan. Add the onions, carrots, celery and fry for 5 minutes. Add the chicken and fry for 7 minutes. Add the courgette and asparagus and fry for a further 3 minutes. Stir in the the crème fraiche and season well with salt and pepper. Transfer the filling to the pie dish.
On a clean, floured surface, unroll the pastry. Butter the rim of the pie dish. Lift the pastry over the top of the pie dish (stretch it gently to fit) and press down around the edges. Trim any excess pastry. Brush the pastry all over with the beaten egg.
Transfer the dish to the oven and bake for 30 minutes, or until the pastry is golden brown and nicely puffed up.