In a bowl, combine 60 ml (¼ cup) of the butter with the flour. Set aside this kneaded butter.
In a second bowl, dissolve the cornstarch in the milk. Set aside.
In a large saucepan, soften the onion, carrot, and celery in the remaining butter. Season with salt and pepper. Add the broth and potatoes and bring to a boil. Cover and simmer for about 10 minutes or until the vegetables are just tender. Add the kneaded butter and boil gently, stirring with a wooden spoon. Add the milk and cornstarch mixture and bring to a boil. Adjust the seasoning. Stir in the chicken and peas. Place plastic wrap directly on the filling. Let cool while preparing the dough.
In a food processor, combine the flour and salt. Add the butter and pulse a few seconds at a time until the butter is the size of peas. Add the water and vinegar, and pulse again until the dough just begins to form. Add water, if needed. Remove the dough from the food processor and form into two discs with your hands. Refrigerate for 30 minutes.
With the rack in the lowest position, preheat the oven to 200 °C (400 °F).
On a floured work surface, roll out the two discs of dough and line a 23-cm (9-inch) pie plate with one of them. Make incisions in the second crust. Spoon the chicken filling into the crust. Brush around the dough with milk and cover with the second sheet of dough. Crimp the edges and brush with milk.
Bake for about 50 minutes or until the pastry is golden brown. Let rest for 15 minutes and serve.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.