Melt the butter in a large skillet over medium heat. Add the diced onion, potato, celery, and carrots. Sauté for about 10 minutes until the vegetables start to soften.

Stir in the sliced mushrooms and cook for another 2 minutes. Sprinkle in the flour and stir for 1 minute to coat all the vegetables evenly.

Slowly pour in the vegetable broth and half-and-half, stirring constantly until the mixture thickens to a creamy consistency. Add the frozen peas and chopped parsley, then season with salt and pepper.

Preheat the oven to 400°F. Spoon the vegetable filling into a large baking dish or pie dish. Unroll the puff pastry over the filling, trimming any excess and pressing the edges down to seal. Cut a few small slits in the top for steam to escape.

In a small bowl, whisk together the egg and milk to make an egg wash, then brush it over the pastry. Bake for 25-35 minutes until the pastry is golden brown. Let cool slightly before serving. Garnish with parsley.