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To make the crust: Combine the flour, buttermilk powder, salt, and baking powder, then mix in the butter and shortening until crumbly, leaving some pea-sized lumps.
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Mix the vinegar with 4 tablespoons of the water. Sprinkle onto the dry ingredients, stir, and squeeze the dough together; if it's not totally cohesive, add an additional tablespoon of water (enough to make the dough stick together nicely, without crumbling).
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Shape the dough into a flattened disk, wrap it in plastic wrap, and refrigerate for 30 minutes.
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To make the filling: In a saucepan set over medium heat, melt the butter, then stir in the flour. Gradually pour in the stock, whisking constantly. Cook and stir the sauce until it comes to a boil, then reduce the heat and simmer for 5 minutes.
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Stir in the chicken and salt. Add ground black pepper, additional salt to taste, and the vegetables.
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To assemble and bake the pie: Spoon the filling into a 9" or 10" deep dish pie pan.
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Roll the crust out slightly larger than the pan. Place it on top of the filling; cut several vent holes, and use any scraps of dough to decorate. Place the pie on a baking sheet (there may be some drips once the filling begins to bubble),
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Bake the pie in a preheated 375°F oven for 40 to 50 minutes, until the crust is golden brown and the filling is bubbly.
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Remove the pie from the oven and let it cool for about 20 minutes before serving. This short rest gives the filling a chance to set, making it easier to cut and plate.
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Store any leftover pie, securely wrapped, in the refrigerator for several days; freeze for longer storage.