Heat the oil in a large skillet over medium-high heat. Add the onions and potatoes and cook until the onions are translucent, about 5 minutes. Add the carrots and mushrooms and cook, stirring occasionally, until the mushrooms are tender and have released their moisture, about 5 minutes. Add the garlic and cook until fragrant, 45 to 60 seconds. Add the flour and cook, stirring, for 1 minute. Add the warmed chicken stock and bring to a boil. Reduce the heat and simmer until thick and bubbly, about 7 minutes. Add the chicken meat and tarragon; taste and adjust the seasoning with salt and pepper. Transfer the chicken mixture to a large casserole dish; set aside to cool slightly.
Heat the oven to 400 degrees F. Remove the pie crust dough from the refrigerator. Dust a clean surface lightly with flour and roll the dough out to form a 12- by 8-inch rectangle, about 1/8-inch thick. Transfer the dough to the top of the casserole dish. Press the dough onto the sides of the dish to seal the edges. Using a paring knife, cut several slits in the top of the crust to allow steam to escape while baking. Bake until golden brown, about 45 minutes. Let stand 10 minutes before serving.
Beginner's Pie Crust:
Combine the flour and salt in the bowl of a food processor fitted with the blade attachment. Add the butter and vegetable shortening and process until the mixture resembles coarse meal, 8 to 10 seconds. Add enough of the ice water, 1 tablespoon at a time, pulsing between additions, until the dough holds together without being sticky or crumbly. Remove the dough from the processor and shape it into a disk. Wrap in plastic wrap. Chill until firm and the moisture has distributed evenly, about 30 minutes.
Dust a clean work surface and a rolling pin with flour. Place the disk of dough in the center of the floured surface. Starting in the center of the dough, roll to, but not over, the upper edge of the dough. Return to the center, and roll down to, but not over, the lower edge. Lift the dough, give it a quarter turn and lay it on the work surface. Continue rolling, repeating the quarter turns, until you have a disk about 1/8 inch thick. Wrap in plastic and refrigerate until ready for use.