Cook the broccoli for 5 minutes until tender. Drain and refresh with cold water, then set aside. Heat the oil in a nonstick frying pan and cook the chicken and bacon over a moderate heat for 7–8 minutes. Add the carrots and cook for a further 3–4 minutes until golden all over. Remove from the heat.
Heat the butter in a medium saucepan and add the flour. Cook over a gentle heat for a few seconds, then remove from the heat and gradually add the milk until well mixed. Add the vinegar and mustard and mix well. Return to the heat and stir continuously until boiled and thickened. Add the crème fraîche and herbs. Add the chicken and vegetables and stir well to coat, then transfer to a round pie dish.
Roll out the pastry on a floured surface to just larger than the dish. Moisten the rim of the dish lightly with a little water, then place the pastry over the top, trim the edges and decorate with any remaining pastry trimmings if liked. Glaze lightly with the beaten egg. Bake in a preheated oven, 180°C (350°F), Gas Mark 4, for 25–30 minutes until crisp and golden.