Place a rack on the top of the oven and preheat to 450 degrees F. Press the dough into a pie plate, prick the bottom of the crust with a fork and flute the edges. Bake the crust until golden, about 10 minutes.
Heat the butter in a large saucepan over medium heat. Cook the onions, stirring occasionally until they are softened, about 5 minutes. Add the garlic, 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper, and cook until fragrant, about 1 minute. Stir in 1 cup of the gravy and the milk, and bring to a boil. Reduce the heat and simmer until thickened, about 5 minutes. Stir in the turkey, carrots, corn, peas, parsley, thyme, oregano, and the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper, making sure all of the ingredients are evenly coated with the gravy. Simmer until the mixture is warmed through, about 5 minutes. Pour the mixture into the pie shell. Use a spatula or wooden spoon to spread the mashed potatoes evenly over the mixture, making sure the potato layer meets the edges of the crust. Cover the potatoes with the cheese. Bake the pie until the top is golden, 25 to 30 minutes.
To serve, cut into slices and drizzle with gravy.
Any leftover vegetables can be used in place of the glazed carrots.