For the Crust:
In a food processor, combine the flour, butter, Parmesan, and salt. Pulse until the mixture resembles fine gravel. Add ¼ to ½ cup water, a few tablespoons at a time, until the dough comes together and forms a ball.
Turn out the dough onto a lightly floured surface and shape it into a disk.
Wrap it in plastic wrap and refrigerate for at least 30 minutes or until ready to use.
For the Roast Chicken:
Preheat the oven to 325°F. Set a roasting rack inside a roasting pan.
In a large bowl, slather the chicken with the olive oil, paprika, salt, pepper, and granulated garlic. Set the chicken on the roasting rack.
Roast until the skin is golden and crispy and a thermometer inserted into the thickest part of the thigh reaches 165°F, 45 to 60 minutes. Set aside to cool.
When cool enough to handle, remove the skin from the thighs. Pull the meat from the bones and tear it into bite-size pieces (discard the bones or use them for stock).
Set aside 4 cups of meat and reserve the rest for later use (chicken salad, BBQ chicken sandwiches, chicken noodle soup).
For the Filling:
In a Dutch oven or large pot, melt the butter over medium-high heat. Add the onion, ½ teaspoon salt, and a dash of pepper, and cook, stirring, until soft, about 3 minutes.
Add the carrots, fennel, and parsnip and cook, stirring, until the vegetables are brightly colored, 2 to 3 minutes.
Stir in the garlic and, when fragrant, stir in the flour. Cook, stirring continuously, for about 5 minutes.
Pour in the sherry and scrape up any bits stuck to the bottom of the
pan. When the liquid has almost evaporated, stir in the broth and milk. Cook until thickened, about 10 minutes, or until the sauce coats the back of a spoon. If the sauce is too thick, add a splash of broth.
Remove from the heat, stir in the reserved 4 cups shredded chicken, the peas, parsley, and thyme. Season with 1 tablespoon salt and ½ teaspoon pepper, or to taste.
Preheat the oven to 350°F. Grease six ramekins or a 9-by-13-inch baking dish with olive oil.
On a lightly floured surface, roll the dough into a ¼-inch-thick rectangle, about 15 inches long and 11 inches wide. If using individual ramekins, cut 6 circles that are 1 inch larger in circumference than the ramekins. If using a single baking dish, cut the dough to that shape, leaving an extra inch around all sides.
Pour the filling into the ramekins or baking dish.
Lay the dough over the filling and trim it to ½ inch from the edge of the dish. Fold the overhang under itself to create a lip around the edge of the dish and use your fingers to flute the edges. Cut a 1-inch vent hole into a small pie or four to six 1-inch vent holes into a large pie.
In a small bowl, lightly beat the egg with 1 tablespoon water. Lightly brush the piecrust(s) with the egg wash.
Bake until the crust is golden brown and the filling is bubbling, about 60 minutes for large pies, 45 to 50 minutes for small pies. Serve immediately.