1. Prepare the Crust:

1. Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
2. Heat olive oil in a skillet over medium heat. Add onion and garlic; sauté until softened, about 3 minutes. Add mushrooms, thyme, salt, and pepper; cook until mushrooms are tender and liquid evaporates, about 8 minutes. 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 diagonal cuts about 1 inch apart along both long sides, leaving a 3-inch uncut center strip for the filling.
4. Spread mushroom mixture evenly down the center strip. Sprinkle Swiss cheese over mushrooms. Fold cut pastry strips over filling, alternating sides to create a braided pattern. Brush with beaten egg.
5. Bake for 20-25 minutes, until pastry is puffed and golden brown. Let cool for 5 minutes, garnish with parsley if desired, slice, and serve warm.