Preparation

• On a lightly floured surface, roll the dough into a circle approximately 13 inches in diameter. Transfer the dough to a standard (not deep dish) 9-1/2-inch pie plate. Gently tuck the pastry into the pan without stretching it, thenfold over the edge and crimp. Refrigerate the pie shell, 30 to 45 minutes.

Make the filling

• Heat the oil in a medium Dutch oven over medium heat. Add the onion, carrot, and celery. Cook, stirring occasionally, until softened, about 5 minutes. Add the beef and garlic, and cook, stirring with a wooden spoon to break up any large pieces of beef, until no longer pink, about 7 minutes.
• Drain off most of the fat in the pan. Return the pan to the heat, stir in the flour and paprika, and cook, stirring, for 1 minute. Add the broth, wine, and tomatoes and their juices, and bring to a simmer. Stir in the tomato paste, chili sauce, Worcestershire, brown sugar, thyme, rosemary, 1/2 tsp. salt, and 1/4 tsp. pepper. Bring to a boil, then reduce to a simmer. Cook, stirring occasionally, until thickened, about 20 minutes. Season to taste with salt and pepper. Remove from the heat and cool completely.
• Position two racks in the oven, one in the bottom third and the other in the center. Heat the oven to 400°F. Place the pie shell on a rimmed baking sheet lined with foil. Spoon in the filling, and smooth with a spoon. Bake on the lower oven rack until the crust begins to brown, about 25 minutes.

Make the topping

• Meanwhile, in a medium saucepan, combine the potatoes with enough well-salted water to cover by a couple of inches. Add the peas. Bring to a boil, then cook at a low boil until the potatoes are tender, about 12 minutes. Drain thoroughly, then transfer back to the saucepan. Add the softened butter, sour cream, and about half of the milk. Using a hand masher, mash thoroughly, adding more milk as needed to make a not-too-loose mixture. (If the mash is too loose, it could spread and fall off the pie.) Season to taste with salt and pepper. Cover the potatoes.
• After 25 minutes, remove the pie from the oven. Spoon a generous mound of the mashed potatoes over the surface. Drizzle with the melted butter, then sprinkle with some of the Parmigiano. Lower the oven temperature to 375°F. Put the sheet on the center oven rack and continue to bake until the cheese is golden and the pie is bubbly, about 35 minutes. Transfer to a rack to cool for at least 25 minutes before serving.