Boil all the potatoes in salted water for 15 mins. Drain and return to pan. Mash with salt and pepper. Stir in the parsley. Set aside.
Put leeks, garlic, carrots, peas and mushrooms with stock for 15 minutes, or until the leeks have softened. Add the cooked chicken and mix in. Add the tarragon, salt and pepper to taste and cook for further 4 minutes.
Transfer the vegetables and chicken to a pie dish. Spread the mash over the mix and brush beaten egg on the top.
Put in the oven for 20 minutes. Serve with green vegetables.