Prepare the Dough:

In a large mixing bowl, combine the flour and salt. Add the cold, cubed butter and use a pastry blender or your fingers to work the butter into the flour until the mixture resembles coarse crumbs.
Mix in the egg and gradually add cold water, stirring until the dough comes together. You might need a bit more or less water. Divide the dough into two parts, one slightly larger than the other. Wrap each in plastic and refrigerate while preparing the filling.

Make the Chicken Filling:

Heat the olive oil in a large skillet over medium heat. Add the chopped onion and minced garlic, sautéing until the onion becomes translucent.
Add the shredded chicken, chicken broth, and cream. Bring to a simmer and let the mixture cook until slightly thickened, about 10 minutes.
Stir in the olives, corn, and peas (if using), season with salt, pepper, and paprika. Remove from heat and stir in the chopped parsley.

Assemble the Pie:

Preheat the oven to 375°F (190°C).
On a floured surface, roll out the larger piece of dough to fit the bottom and sides of a pie dish. Press the dough into the dish.
Spoon the filling into the crust.
Roll out the second piece of dough and place over the filling. Seal the edges by crimping and trim any excess dough.
Make a few slits on the top crust to allow steam to escape.

Bake:

Beat an additional egg and brush the top crust for a golden finish.
Bake in the preheated oven for about 35-40 minutes, or until the crust is golden brown.

Serve:

Let the pie cool slightly before cutting into it. Serve warm.