Preheat oven to 350 degrees f/180 degrees c. Line a large baking tray with parchment paper and set aside.
In a large skillet set over medium to high heat, add the olive oil and onions. Stir well and cook down until soft, about 5 minutes.
Add the mushrooms and garlic to the onions and give a good stir. Cook down until the mushrooms are soft and browned. About another 5 - 8 minutes.
Reduce heat to low and add the cream cheese, parmesan cheese, oregano, salt, pepper and parsley. Stir really well to completely combine and you should have a soft and creamy texture. Remove from heat.
Add one of the whisked eggs to the mixture and stir quickly into the creamy mixture until it's completely incorporated.
Remove the puff pastry from the fridge and unroll it. Lay the pastry sheet on a sheet of parchment paper. Think of the pastry in thirds. Cut approximately 8 equal diagonal strips on each side. Start the shaping of the pastry by cutting the top corners off and then making a small oval cut at the top.
Spread the mushroom filling down the centre, leaving a 1/2 inch (1 cm) space at the top and the bottom, fold the pastry over the filling. Fold the pastry strips over each other. Gently press down the edges so they stay shut.
Lift the parchment paper onto the baking tray. Brush the top of the pastry braid with the remaining whisked egg. Bake between 20 - 22 minutes until golden and puffed up.
Remove from oven and leave to cool for about 10 minutes before cutting. Cut into slices and serve hot.