Preheat the oven to 400°F Heat the oil and butter in a medium-size saucepan over medium-low heat.
Add the onions and turn the heat to medium-high, then cook until they begin to caramelize and turn brown, 6 to 8 minutes.
Add the garlic and stir for 10 seconds, then add the porcini mushrooms and cook for 3 minutes.
Add the thyme, Sherry, cheese, lemon juice, lemon zest and nutmeg. Cook for 2 minutes, then add the vinegar and remove from the heat.
Place the arugula in a medium-size bowl and pour the mixture over it. Season with salt and pepper, and add the parsley. Stir to combine.
Place the puff pastry sheets on baking sheets (there is no need to roll them out). Pierce the pastry sheets with the tines of a fork, then brush the tops with a beaten egg. Parbake the puff pastry for 13 minutes.
Remove from the oven and distribute the filling evenly over the pastry sheets. Sprinkle a bit of cheese and season with a little black pepper.
Bake for 6 minutes. Remove from the oven, let cook, and serve at room temperature.