Preheat the oven to 400°F (200°C).
On a well-floured surface, roll refrigerated pie dough to about 12 inches in diameter and fit it into a 9-inch deep pie dish, pressing firmly into the bottom and sides; fold or crimp the overhang.
Prick the bottom and sides of the crust all over with a fork to prevent bubbling, then transfer the pie dish to a baking sheet.
Par-bake the crust on the baking sheet for 15 to 18 minutes, until lightly set and just beginning to color, then remove from oven.
In a large bowl, lightly beat the 7 eggs with 2/3 cup skim milk until combined, then stir in the diced cooked chicken, chopped spinach, Swiss cheese, mushrooms, salt, black pepper, and dried thyme.
Pour the filling into the par-baked crust; place the pie on the baking sheet and loosely cover the exposed edges of the crust with foil to prevent over-browning.
Bake on the middle rack for 50 to 60 minutes, or until the eggs are fully set and a knife inserted into the center comes out clean.
Remove the quiche from the oven and let it cool before slicing and serving.