In a small bowl, combine 4 tablespoons of the softened butter with the 1/4 cup of all-purpose flour to create a smooth mixture. Set this aside.

In another bowl, dissolve the cornstarch in the milk and set aside.

In a large saucepan, melt the remaining 2 tablespoons of butter over medium-high heat. Sauté the onion, carrot, and celery for about 3 minutes until tender.

Pour in the chicken broth or your water with bouillon cubes, add the diced potatoes, and bring to a boil. Cover and simmer for around 10 minutes until the vegetables are tender.

Stir in the butter and flour mix to thicken, then add the milk and cornstarch mixture, stirring until thickened. Season with salt and pepper to taste, then incorporate the chicken and peas.

In a food processor, combine 2 cups of flour and 1/2 tsp salt. Add cold butter cubes and pulse until the mixture resembles coarse crumbs.

Add the ice water and vinegar, pulsing until just combined but still crumbly.

Transfer the dough onto Saran wrap, shape into a ball, divide into two disks, flatten, and refrigerate for one hour.

Preheat your oven to 400°F (200°C). Roll out the larger disk to fit a 9-inch pie dish, letting it overhang slightly, and roll the smaller disk to about 10 inches in diameter.

Cut slits in the smaller dough for steam to escape, spoon filling into the pie dish, cover with the second layer of dough, crimp edges, and brush with egg wash.

Bake for about 50 minutes until the pastry is golden brown. Allow the pie to rest for 15 minutes before serving.