1. Prepare the Crust:

1. Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
2. In a large skillet over medium heat, melt butter with olive oil. Add sliced onion and cook, stirring occasionally, until softened and golden brown, about 10-12 minutes. Transfer to a bowl.
3. In the same skillet, add mushrooms and cook over medium-high heat until they release their liquid and start to brown, about 7-8 minutes. Add garlic, thyme, salt, and pepper, and cook for 1 minute more. Remove from heat and let cool slightly.
4. On a lightly floured surface, roll out the puff pastry sheet to a 10×12-inch rectangle. Transfer to the prepared baking sheet. Spread the caramelized onions and mushroom mixture evenly down the center third of the pastry, leaving a 1-inch border at the top and bottom. Sprinkle the Gruyère cheese over the top.
5. Using a sharp knife or pizza cutter, make diagonal cuts about 1 inch apart along both long sides of the pastry, cutting from the edge to the filling. Fold the strips over the filling in an alternating pattern to create a braid. Tuck the ends under to seal.
6. In a small bowl, whisk together the beaten egg and water. Brush the egg wash over the entire pastry braid. Sprinkle with sesame seeds or poppy seeds if desired.
7. Bake for 20-25 minutes, until the pastry is golden brown and puffed. Let cool on the baking sheet for 5 minutes before slicing. Serve warm or at room temperature.