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

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

Add minced garlic and sliced cremini mushrooms, cooking for about 8 minutes until mushrooms are browned and moisture is released.

Sprinkle in thyme, salt, and pepper. If using white wine, add it now and cook until liquid is absorbed.

Remove from heat and stir in cream cheese, Parmesan, and Dijon mustard. Mix smoothly and fold in Gruyère cheese.

Once cooled, incorporate one beaten egg and mix thoroughly.

Roll out the puff pastry on a lightly floured surface. Cut diagonal strips along both sides.

Spoon the filling into the center section of the pastry, then fold the cut strips over the filling in an alternating pattern.

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

Bake for 20-22 minutes until deep golden brown and crisp.

Let the braid cool for about 10 minutes before slicing and serving warm.