In a large bowl, whisk together the flour and salt. Cut in the cold butter using a pastry blender or your fingers until the mixture resembles coarse crumbs. Gradually add the ice water, one tablespoon at a time, mixing until the dough just comes together. Form the dough into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Heat the olive oil in a large saucepan over medium heat. Add the onion, carrots, and celery and cook until softened, about 5-7 minutes. Add the mushrooms and cook until they release their liquid and start to brown, about 5 minutes more. Stir in the flour, thyme, salt, and pepper. Gradually whisk in the vegetable broth and bring to a simmer, stirring constantly until the sauce thickens. Stir in the peas, corn, and parsley.

Preheat oven to 400°F (200°C). On a lightly floured surface, roll out the dough into a circle slightly larger than your pie dish. Transfer the dough to the pie dish and trim any excess dough. Pour the vegetable filling into the crust. If desired, brush the crust with beaten egg for a golden finish. Cut a few slits in the top of the crust to allow steam to escape. Bake for 35-45 minutes, or until the crust is golden brown and the filling is bubbly. Let cool for 10-15 minutes before serving.