In a large skillet, heat olive oil over medium heat. Add sliced mushrooms and cook until they are browned and their moisture has evaporated, about 8-10 minutes.

Add minced garlic, thyme, salt, and pepper to the skillet. Cook for an additional 2 minutes, then remove from heat and let cool slightly.

Preheat the oven to 400°F (200°C). Roll out the puff pastry sheet on a lightly floured surface to smooth out creases.

Place the cooled mushroom mixture in the center of the pastry, then sprinkle shredded Gruyère cheese on top.

Cut slits along the sides of the pastry, about 1 inch apart, to create flaps. Fold the flaps over the filling, alternating sides to create a braid effect.

Brush the top of the braid with the beaten egg to give it a golden color when baked.

Transfer the braided pastry to a baking sheet lined with parchment paper. Bake in the preheated oven for 25-30 minutes or until golden brown and puffed.

Remove from the oven and let cool for a few minutes before slicing and serving.