Prepare Old-Fashioned Flaky Pie Dough according to the recipe. After rolling, folding, and dividing dough in half, roll one portion into a 14-inch round. Transfer to a parchment-lined aluminum rimmed baking sheet, cover with plastic, and refrigerate to relax and chill dough, at least 2 hours or up to 24.
In a large sauté pan or cast iron skillet, heat 4 tablespoons (60ml) olive oil over medium-high heat until shimmering. Add mushrooms and cook, stirring occasionally, until browned, about 6 minutes.
Stir in asparagus stalks, thyme, and the remaining 1 tablespoon (15ml) olive oil. Lower heat to medium and cook, tossing and stirring, until asparagus is tender, about 8 minutes. Add 3 tablespoons (45ml) water to skillet, scraping up browned bits from bottom of pan. Season with salt and remove from heat.
Meanwhile, in a large saucepan, melt butter over medium heat until foaming. Add leeks and cook, stirring, until very soft, about 8 minutes. Lower heat if necessary to prevent browning. Season with salt.
Add the leeks to the asparagus mixture along with the cheese and stir to combine.
