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

In a large skillet, heat olive oil and unsalted butter over medium heat. Add diced yellow onion and cook for about 5 minutes until soft.

Add minced garlic and sliced cremini mushrooms to the skillet. Sauté for approximately 8 minutes until browned.

Season the mixture with fresh thyme, salt, and pepper. If desired, add a splash of white wine or broth, cooking for an additional minute.

Lower the heat, stir in softened cream cheese, grated Parmesan, and optional Dijon mustard until smooth. Fold in grated Gruyère cheese.

Once the filling has cooled slightly, stir in the beaten egg to bind the mixture.

Unroll the puff pastry on a lightly floured surface and make diagonal cuts along each side, creating strips for the braid. Lay the filling down the center.

Fold the ends of the pastry over the filling, then alternate folding the strips over the filling to create a braid.

Transfer the braided pastry to the prepared baking tray. Brush the top with remaining beaten egg and bake for 20-22 minutes until golden brown.

Remove from the oven and allow to cool for about 10 minutes. Garnish with fresh parsley and serve warm.