For the crust: Combine the flour, butter, Parmesan and a pinch of salt in the bowl of a food processor fitted with the pastry blade. Pulse a few times until the mixture resembles fine gravel. Add up to 8 tablespoons of ice water, a few tablespoons at a time, until the dough comes together and forms a ball. Remove from the food processor and form the dough into a disk; wrap it in plastic wrap and refrigerate for at least 30 minutes or until ready to use.
For the roast chicken: Meanwhile, preheat the oven to 325 degrees F. In a large bowl, slather the chicken with the olive oil and sprinkle with the granulated garlic, paprika, 1 teaspoon salt and 1/2 teaspoon pepper. Set the chicken on a rack in a roasting pan and roast until the skin is golden and crispy and a thermometer inserted into the thickest part of a thigh reaches 165 degrees F, 45 to 60 minutes. Set aside to cool.
When cool enough to handle, remove the skin, pull the meat from the bones and tear it into bite-size pieces (you need 4 cups of the meat for the pie).
For the filling: Heat the butter in a Dutch oven over medium-high heat. When the butter has melted, add the onion, 1/2 teaspoon salt and a dash of pepper and cook until soft, about 3 minutes. Add the carrots, fennel and parsnip and cook until the vegetables are brightly colored, 2 to 3 minutes. Add the garlic and when fragrant, stir in the flour. Cook for about 5 minutes, stirring constantly. Add the sherry and scrape up any bits stuck to the pan. When the liquid has almost evaporated, stir in the chicken stock and milk and cook, stirring, until the sauce is thick enough to coat the back of a spoon, about 10 minutes. If the sauce becomes too thick, add more chicken stock to reach the desired consistency. Remove from the heat and stir in the chicken, peas, parsley, thyme, 1 tablespoon salt and 1/2 teaspoon pepper.
To assemble: Preheat the oven to 350 degrees F. On a lightly floured surface, roll the dough to a 1/4-inch-thick rectangle, about 15 inches long by 11 inches wide. If using individual ramekins, cut 6 circles in the shape of the ramekins plus an extra 1-inch overhang. If using a 9-by-13-inch pan or similar-sized shallow baking dish, cut to that shape plus an extra 1-inch overhang.
Pour the filling into the ramekins or pan. Lay the dough over the filling and trim to 1/2 inch from the edge. Fold the overhang under itself to create a lip around the edge and use your fingers to flute the edges, or use the tines of a fork to create an attractive edge. Alternatively, simply tuck the overhang dough into the side of the pan. Cut a 1-inch vent hole in each small pie or four to six 1-inch vent holes in a large pie.
In a small bowl, lightly beat the egg with 1 tablespoon water. Lightly brush the pie crust with the beaten egg.
Bake until the crust is golden brown and the filling is bubbly, 45 to 50 minutes for small pies and about 60 minutes for a large pie.
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