Turn the dough onto a lightly floured counter or work surface. Pack it together, then divide in half. Knead one-half of it two or three times, then flatten into a 1⁄2-inch-thick disk. Wrap in plastic wrap and refrigerate for 1 to 2 hours. Repeat with the other half, then wrap, refrigerate, and use within 2 days (or place in a plastic freezer bag and freeze for up to 2 months).
Melt the butter in a large saucepan over medium-low heat. Add the onion and celery, stir, cover, and cook for 6 to 7 minutes, or until soft, stirring occasionally. Add the flour, increase the heat to medium, and cook for 1 minute, stirring. Add the stock and whisk to blend. As the sauce starts to thicken, add the milk, sage, and thyme and whisk to blend. Add the chicken and vegetables and continue to cook for 3 to 4 minutes, or until heated through, stirring. Add 1⁄3 cup of the Parmesan cheese and remove from the heat. Add salt and pepper, to taste.
Butter a shallow 2- to 2-1⁄2-quart baking dish. Transfer the filling to the dish and smooth the top. Sprinkle with the remaining Parmesan cheese. Set aside for 15 minutes.
Preheat the oven to 400°F.
On a sheet of wax paper, roll the pastry to be slightly larger than the dish. Invert the pastry over the filling and peel off the paper. Tuck the pastry edges into the dish. With a paring knife, make several cuts, or steam vents, in the pastry. Brush the pastry with the egg glaze. (If the baking dish is quite full, place it on an aluminum foil–lined baking sheet to catch spillovers.)
Bake on the center oven rack for 40 minutes, or until golden and bubbling.