Preheat your oven to 350°F (180°C) and prepare a baking tray with parchment paper.

Heat a skillet over medium heat and add olive oil and butter. Sauté onion until translucent, about 5 minutes.

Add mushrooms and sauté until browned, about 8 minutes. Stir in garlic, thyme, salt, and pepper.

Pour in white wine (if using) and cook until absorbed. Then mix in cream cheese, Parmesan, and Gruyère, allowing to cool slightly.

Roll out puff pastry on a floured surface and cut diagonal strips on both sides of the center.

Place the mushroom mixture down the center, fold pastry strips over to braid, and seal edges.

Brush with egg wash and bake for 20-22 minutes until golden brown.

Cool for a few minutes, slice, and serve warm.