For the crust:

1. In a medium-sized bowl, mix together the flour and salt. Dice the butter into small pieces and crumble it with the flour until lumps are the size of small peas. Add sugar and stir well. Pour in the ice cold water, little at a time, mixing with a fork. Finish by hand and shape into a ball.
2. Flatten the dough a little bit, wrap into cling film and chill in the refrigerator for about 2 hours.

For the filling:

1. Preheat the oven to 425°F (220°C). Grease the bottom and sides of a 9-inch (23 cm) pie dish with butter and dust with flour.
2. Place the carrots and potatoes in a pan with just enough water to cover them. Heat over medium heat until soft, about 15 minutes. Drain all the vegetables. Set aside.
3. Melt butter in a large skillet placed over medium heat. Add the onions, garlic and celery and cook for about 3 minutes, until tender. Add mushrooms and cook for one more minute. Whisk in the flour, salt, pepper, nutmeg and thyme. Slowly add the broth and cream, stirring frequently until thick, about 10 minutes. Turn off the heat and add the peas, and stir in the remaining veggies.
4. Roll out the first pie crust into a large circle, about 12-inch (30 cm) in diameter. Carefully place over the pie dish, prick with a fork and pour the filling over.
5. Roll out the second pie crust and use it to cover the pie. Trim the extra overhang off the sides. Seal the edges by crimping with your fingers. With a small sharp knife, make a few slits in the top crust for steam to escape.
6. Brush with beaten egg and bake for about 30 minutes or until crust is golden. Check the pot pie after 15 minutes, and place a piece of foil loosely over the top if the top starts to brown, then continue baking. Allow to cool slightly and serve!