TO MAKE THE BISCUIT DOUGH

Combine the flour, baking soda, baking powder, salt, and pepper in a large bowl. Stir to mix well. Use a pastry cutter, two forks, or your fingertips to work the butter into the flour until the mixture resembles coarse meal. Stir in the buttermilk, and blend just until the dough holds together. Pat the dough into a ½-inch thick round, wrap in plastic wrap, and refrigerate while you are making the filling.

TO MAKE THE FILLING

Preheat the oven to 400°F. Have ready an 8-cup baking dish about 2 inches deep, or use a 10-inch cast-iron skillet with 2-inch sides. In a 1-quart saucepan, bring the stock to a boil. Add the carrot and cook until crisp-tender, about 10 minutes. Using a slotted spoon, transfer the carrot to a plate and set aside. Turn off the heat under the stock.

In a 10-inch skillet, heat the butter with the oil over medium heat until the butter foams. Add the onion and sauté until it begins to soften, about 2 minutes. Add the mushrooms and sauté until they just begin to brown, about 3 minutes longer. Sprinkle the flour over the onion-mushroom mixture and stir to blend in. Slowly stir in the stock, bring to a simmer, and stir until smooth and thickened, about 2 minutes. Add the cream, stir to blend, and bring to a simmer. Add the reserved carrot, the turkey, and parsley, and stir to combine. Return the mixture to a simmer; then add salt and pepper to taste. Remove from the heat. Transfer the filling to a baking pan, unless you have cooked the filling in a cast-iron skillet.

Turn out the biscuit dough onto a floured work surface. Flour a rolling pin, and roll out the dough into a 10-inch circle. Carefully place the dough over the filling, centering it in the pan. Lightly press the edges of the dough against the sides of the pan. Brush the top of the dough with the milk. Cut 3 slits, each 2 inches long, in the center of the dough. Bake until the dough is nicely browned, about 25 minutes. Serve immediately.