Combine the flour, salt, and butter in a large mixing bowl.
Cut and mix the butter into the flour with a pastry cutter until the mixture has a crumbly appearance.
Stir in the water, 1 tablespoon at a time, until a uniform dough forms, and you can still see a few chunks of butter. You may not need to use all of the water. Do not overmix.
Set in the fridge to chill while preparing the filling.
For the Filling:
Heat the butter in a large, deep skillet over medium heat. Add the onion, potatoes, carrots, garlic, and curry powder. Cook for about 5 minutes.
Add the chicken, broth, mushrooms, peas, salt, and pepper. Bring to a boil, then simmer for 10 minutes, stirring occasionally.
In a small bowl, combine the cornstarch and water to create a slurry. Mix into the filling and cook for another minute or two, until the liquid thickens. Remove from the heat and let cool slightly.
To Assemble and Bake:
Preheat the oven to 400°F. While the filling is cooling, set out a 9-inch deep dish pie pan.
Remove the pie dough from the fridge and place on a well-floured surface. Divide it equally in half, and roll out one half to 1/8-inch thickness using a rolling pin. Measure it to fit the base of the pie pan, being sure there is a little excess around the edges. Carefully transfer it to the bottom of the pie pan, pressing gently down on the bottom and around the sides, leaving the overhang. Place it back in the refrigerator for about 10-15 minutes, until cold and firm.
Remove the pan from the refrigerator. Spoon all of the filling into it.
Roll out the remaining dough to 1/8-inch thickness. Measure it to fit the top of the pan, with a little excess around the edges. Gently place it over the filling. Fold the overhang underneath itself, and crimp the edges with your fingers or a fork. Cut a few slits in the top of the dough.
In a small bowl, whisk together the egg and water for the egg wash. Generously brush the egg wash over the top of the pie crust.
Bake for 30-40 minutes, or until the top is golden brown and the filling is bubbling underneath the slits. Remove from the oven and let cool for at least 10 minutes before serving.