Step 1: Make the Crust

1. In a large mixing bowl, combine the flour and salt.
2. Cut in the shortening with a pastry cutter or fork until the mixture resembles coarse crumbs.
3. In a separate bowl, whisk together the beaten egg, ice water, and vinegar. Gradually add this wet mixture to the dry ingredients and stir until just combined.
4. Divide the dough into two equal portions. Shape each into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Step 2: Prepare the Filling

1. Melt the butter in a large skillet over medium heat.
2. Add the diced vegetables and cook until softened, about 5 minutes.
3. Stir in the flour and cook for 1–2 minutes, whisking to remove the raw taste.
4. Gradually add the chicken stock while stirring, followed by the half-and-half. Stir until the sauce thickens.
5. Add the chopped chicken, salt, and pepper. Mix well and set aside.

Step 3: Assemble the Pot Pie

1. Preheat your oven to 375°F (190°C).
2. Roll out one disk of chilled dough on a floured surface and fit it into a 9-inch pie dish. Trim the edges if needed.
3. Pour the prepared filling into the crust, spreading it evenly.
4. Roll out the second disk of dough and place it over the filling. Seal and crimp the edges. Cut small slits in the top crust to allow steam to escape.

Step 4: Bake

1. Place the pie on a baking sheet to catch any drips.
2. Bake for 35–40 minutes, or until the crust is golden brown and the filling is bubbling.
3. Let the pie cool for 10 minutes before slicing and serving.