Heat oil and butter in a frying pan, add chicken mince and garlic. Stir over medium heat until chicken is cooked and liquid has evaporated. Use a fork to break up any lumps as mince cooks.
Add leek and cook over low heat for 2–3 minutes or until softened. Add the flour and cook for 1 minute. Add wine and bring to a boil for 1 minute, then pour in stock and cream and cook, stirring, for a further 5 minutes over medium heat. Stir through the cooked vegetables and tarragon, season with salt and pepper. Set aside to cool.
Invert six individual pie tins over pastry, cut around tin edge for the lids of pies. Set aside. For the pie bases, invert tins over remaining pastry and cut circles 1.5cm bigger than the tin. Lightly grease tins, line base with pastry, fill with chicken mixture. Place pastry lids over and seal edges with a fork.
Place pies on oven tray and brush with beaten egg. Cook in 200°C pre-heated oven for 25 minutes until golden brown.