For the pastry, pulse the flour, baking powder, salt, butter and egg yolk in a food processor until the mixture resembles breadcrumbs.
Gradually add the water, a tablespoon at a time, mixing continuously until the mixture just comes together as a dough. (You may not need to use all the water.) Roll the dough into a ball, then wrap it in cling film and chill in the fridge for an hour.
Meanwhile, for the filling, heat two tablespoons of the olive oil in a saucepan over a medium heat, add the chicken pieces and cook for 4-5 minutes, or until browned. Add the wine, water, tarragon and thyme, and season, to taste, with salt and freshly ground black pepper. Bring the mixture to the boil, reduce the heat, and simmer for 15-20 minutes, or until the volume of liquid has reduced by half and the chicken is cooked through.
Heat the remaining olive oil in a separate frying pan over a medium heat, add the mushrooms, and cook for 4-5 minutes, or until golden-brown, then season, to taste, with salt and freshly ground black pepper.
Remove the cooked chicken from the frying pan with a slotted spoon and place into a bowl, then add two spoonfuls of the residual juices from that pan. Add the mushrooms, then stir in the soured cream until well combined. Set aside to cool.
Preheat the oven to 180C/350F/Gas 4. Grease a deep 20cm/8in pie tin with butter. Add a small amount of flour, then turn the tin to coat the base and sides, shaking out any excess.
When the pastry has chilled, roll out two thirds of it onto a clean, floured work surface, until it is almost twice as wide as the diameter of the tin. Using the rolling pin, lift the pastry and lay it over the pie tin to line the base and sides. Gently press the pastry into the corners of the tin, trimming off any excess, then prick the base of the pastry case several times with a fork.
Cover the pastry with a sheet of greaseproof paper and fill it with rice or dried beans. Bake in the oven for 10-15 minutes, or until the pastry is pale golden-brown. Remove from the oven, discard the beans and greaseproof paper and set the pastry aside to cool. When cool, pour the pie filling into the pastry case.
Roll the remaining one third of the pastry out onto a clean, floured work surface until it is a little bit larger than the diameter of the pie tin. Brush the rim of the cooked pastry case with some of the beaten egg and place the pastry lid on top of the pie. Trim off any excess pastry. Seal the pastry lid to the pastry case by crimping the edges of the pastry lid with a fork.
With a knife, make two small holes in the centre of the pastry lid to allow steam to escape. Brush the pastry lid with the remaining beaten egg, then transfer to the oven and bake for 30-35 minutes, or until golden-brown.