Roll the pastry out to 2cm thickness on a lightly floured surface and cut into 4 equal rounds with diameter of 10cm each. Reserve the left over pastry and gather into a ball.
Use to line the base and sides of 5cm metal ring moulds with a depth of 6-7cm. Carefully transfer to a lined baking tray, prick the bases and chill until needed.
Meanwhile, heat the olive oil in a large frying pan over a moderate heat and saute the shallots for 4-5 minutes, stirring occasionally until softened. Add the chicken, ground tarragon and some seasoning and continue to cook for 6-8 minutes, stirring occasionally, until the chicken is just cooked with no pink showing.
Remove from the heat and lay the mixture out on a chopping board to cool. Roughly chop the chicken pieces once they have cooled a little and gather the mixture up into a mixing bowl. Add the chopped tarragon and mix well.
Preheat the oven to 190°C. Remove the pastry from the fridge and spoon into the cases, filling them all the way to the top. Roll the reserved pastry out on a lightly floured surface and cut 4 rounds that will act as lids for the pies. They should not overlap the pastry rim, rather, sit flush next to the pastry at the rim, to give an appearance similar to the top of a brioche bun.
Brush the tops with the beaten egg and bake in the oven for 25-30 minutes until the pastry is golden and cooked and the filling is hot. Remove and arrange on saucers before serving.
For top tips on protecting you and your family when preparing raw meat and poultry, visit Food Safety in the Home and view the full Food safety at home video.