1. Prepare the Crust:

1. Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
2. In a skillet, heat olive oil over medium heat. Add onion and cook until softened and caramelized, about 10 minutes. Add mushrooms and cook until browned and liquid evaporates, about 8 minutes. Stir in garlic, thyme, salt, and pepper. Remove from heat and let cool slightly.
3. Roll out puff pastry on a lightly floured surface into a 10×12-inch rectangle. Transfer to prepared baking sheet. Using a sharp knife, make 1-inch diagonal cuts along both long sides, leaving a 3-inch uncut center strip for filling.
4. Spread mushroom mixture down the center strip, then top evenly with shredded Gruyère. Fold cut strips over the filling in a crisscross braid pattern, alternating sides and pressing ends to seal.
5. Brush braid with beaten egg and sprinkle with flaky sea salt. Bake for 20-25 minutes until pastry is puffed and golden brown. Let cool for 5 minutes before slicing.