Remove chunks of cooked chicken from the bone, removing any skin.
In a large pan at medium-high heat, melt the butter and add the spring onions, tarragon and chives. Lightly brown the spring onions, add the tablespoon of flour along with 2 tablespoons of cold water and stir until it becomes a smooth paste. Add the milk and the cream, and cook until slightly thickened, about 10 minutes.
Stir in the chicken and the mushrooms. Season with salt and pepper.
Pour the filling into a medium sized pie dish (it should be large enough to feed 2 to 3 people). Roll out puff pastry and lay on top of the pie dish. Cut any excess from the edges.
Whisk an egg in a small bowl and brush on top of the pastry.
Bake in the preheated oven until the pastry is crispy, 30 to 35 minutes.