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In a bowl, mix 1/4 cup (60 ml) softened butter with flour. Set aside.
In a large saucepan, cook carrots, potato, leek, and celery in 2 tbs. (30 ml) butter until softened. Add salt and pepper. Add stock. Cover and let simmer about 10 minutes or until the vegetables are just tender.
Add butter/flour mixture and bring to boil, stirring gently with a wooden spoon. Add cream and bring to boil once more. If necessary, adjust seasoning.
Incorporate chicken, herbs, and peas.
Place plastic wrap directly on the filling to prevent it from drying. Let cool while you prepare the crust.
In a food processor, mix flour and salt. Add butter and pulse mixture until the butter is broken up to pea-sized pieces.
Add water and vinegar and mix again just until the pastry starts to take shape. Add water if necessary.
Remove pastry from food processor and shape it into two discs with your hands. Refrigerate 30 minutes.
Place the oven rack in the bottom position. Preheat oven to 400°F (200°C).
On a floured worktop, roll out the two pastry discs and press one into the bottom of a 9 in. (23 cm) pie dish. Cut slits in the second layer of pastry.
Spread the chicken filling over the pastry. Brush edges of the pastry with milk and cover with the second layer of pastry. Scallop the edges and brush with milk.
Bake about 50 minutes or until pastry is golden. Let rest 15 minutes and serve.
Use leftover roast chicken or turkey for this recipe.