Heat olive oil in a large skillet over medium heat. Add finely chopped onion and sauté until translucent, about 3-4 minutes.

Add minced garlic and sliced mushrooms, cooking until mushrooms are tender, roughly 5-7 minutes. Stir in thyme, salt, and pepper; let cool.

Roll out puff pastry into a 12x16 inch rectangle on a floured surface. Transfer to a parchment-lined baking sheet.

Make diagonal cuts on each side of the pastry, about 1 inch apart, leaving the center third uncut for filling.

Spread the cooled mushroom mixture down the center of the pastry. Top with shredded Gruyère cheese.

Fold the cut edges over the filling, alternating sides to create a braid. Pinch the ends to seal.

Preheat oven to 400°F (200°C). Brush the braid with a beaten egg. Bake for 25-30 minutes until puffed and golden brown.

Allow to cool for 5-10 minutes before slicing and serving warm.