Preheat the oven to 400 degrees F. Grease a 9-inch soufflé dish.
In a large sauté pan, melt the butter. Add the onions, mushrooms and celery, and sauté for 2 minutes. Season with salt and pepper. Stir in the flour and cook for about 3 to 4 minutes for a blond roux. Stir in the vegetable stock and bring the liquid up to a boil. Reduce to a simmer and continue to cook for 4 to 6 minutes, or until the sauce starts to thicken. Stir in the half-and-half and continue to cook for 4 minutes. Season with salt and pepper. Stir in the potatoes, carrots, peas and broccoli. Mix the filling thoroughly. Pour the filling into the soufflé dish. Place the top crust on top of the filling. Carefully tuck the overlapping crust into the pan, forming a thick edge. Crimp the edges of the pan and place on a baking sheet. Brush with egg wash. Slice two slits into the crust as steam vents. Place the sheet in the oven and bake for about 25 to 30 minutes or until the crust is golden brown and crispy.
Remove from the oven and cool for 5 minutes before serving.