01 - Combine flour and salt in a large bowl. Cut in cold butter until mixture resembles coarse crumbs. Gradually add ice water, mixing until dough just forms. Shape into a disk, wrap tightly, and refrigerate for 30 minutes.
02 - Heat olive oil and butter in a skillet over medium heat. Add sliced onions and cook, stirring occasionally, until golden and caramelized, about 15 minutes.
03 - Stir in minced garlic and cook for 1 minute. Add mushrooms, thyme, salt, and pepper. Sauté until mushrooms release liquid and brown, about 8 to 10 minutes. Remove pan from heat and stir in chopped parsley. Let mixture cool slightly.
04 - Preheat oven to 400°F. On a floured surface, roll chilled dough into a 12-inch circle. Transfer to a parchment-lined baking sheet.
05 - Sprinkle half of the grated cheese onto the center of the dough, leaving a 2-inch border. Spread the mushroom and onion filling over the cheese. Top with remaining cheese.
06 - Fold dough edges up over filling, pleating as needed to form a rustic border. Brush crust with beaten egg to glaze.
07 - Bake for 30 to 35 minutes until the crust is golden and crisp. Allow to cool slightly before slicing.
08 - Optionally sprinkle flaky sea salt and extra parsley on top before serving.