01 - Whisk together flour and salt in a large bowl. Cut in cold butter until mixture resembles coarse crumbs. Gradually add ice water, mixing until dough just comes together. Shape into disk, wrap in plastic, and chill 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 - Add minced garlic and cook 1 minute. Stir in mushrooms, thyme, salt, and pepper. Cook until mushrooms release moisture and begin browning, about 8-10 minutes. Remove from heat and stir in parsley, then let cool slightly.
04 - Preheat oven to 400°F. On floured surface, roll dough into a 12-inch circle. Transfer to parchment-lined baking sheet.
05 - Sprinkle half the grated cheese over center of dough, leaving a 2-inch border. Spread mushroom and onion mixture over cheese and sprinkle with remaining cheese. Fold edges of dough up over filling, pleating as needed. Brush crust with beaten egg.
06 - Bake for 30 to 35 minutes or until crust is golden brown and crisp. Allow to cool slightly before slicing.
07 - Optional: Sprinkle with flaky sea salt and additional parsley before serving.