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

Heat a tablespoon of olive oil and a tablespoon of butter in a large skillet over medium heat until melted. Add the chopped onion and cook for about 5 minutes until translucent.

Add minced garlic and cremini mushrooms to the skillet, cooking for about 8 minutes until browned.

Season with thyme, salt, and pepper. If using, add white wine and cook for about 3 minutes until mostly evaporated.

Mix in cream cheese, Parmesan, and Dijon mustard on low heat until melted, then fold in Gruyère cheese.

Allow the filling to cool for a few minutes and mix in one beaten egg to bind.

Unroll chilled puff pastry on a floured surface and cut diagonal strips along the edges, leaving the center intact.

Center the cooled filling over the middle section of the pastry, fold the edges over, and braid the diagonal strips across the filling.

Brush the braided pastry with a beaten egg for a golden finish.

Bake for 20-22 minutes until golden and flaky.

Let the pastry cool on the tray for about 10 minutes before serving.