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

In a skillet, warm the olive oil and butter over medium heat. Sauté the diced onion for about 5 minutes until soft.

Add minced garlic and sliced mushrooms, cooking for approximately 8 minutes until mushrooms are browned.

Sprinkle in thyme, and season with salt and pepper. Optionally add a splash of white wine and cook until absorbed.

Lower the heat, mix in cream cheese and Parmesan cheese until smooth, then fold in Gruyère cheese.

Remove from heat, cool slightly, then stir in a beaten egg to bind the filling.

Roll out the puff pastry on a floured surface to about ¼ inch thickness, and cut diagonal strips along the edges.

Spoon the mushroom filling into the center third of the pastry, fold ends over, and braid the strips alternately.

Transfer the braid to the baking tray and glaze the top with a beaten egg.

Bake for 20-22 minutes until puffed and golden brown.

Let the braid rest for 10 minutes before serving warm.