Make the pie crust by combining 2 1/2 cups flour and 1 teaspoon salt in a large bowl. Use a pastry cutter or your fingers to work the cold, cubed butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces remaining.
Drizzle 6 tablespoons of ice water over the flour mixture, stirring with a fork until the dough just begins to clump together. If the dough is too dry, add the remaining water 1 tablespoon at a time. Gather the dough into a ball, divide it in half, flatten each half into a 1-inch thick disk, wrap in plastic, and refrigerate for at least 1 hour.
While the dough chills, prepare the filling. Heat the olive oil in a large skillet over medium-high heat. Add the chicken pieces and cook for 5 to 7 minutes, stirring occasionally, until browned and cooked through. Use a slotted spoon to transfer the cooked chicken to a clean bowl.
Add the sliced leeks to the same skillet and cook for 5 minutes over medium heat until softened. Add the minced garlic and cook for 1 more minute until fragrant.
Sprinkle 3 tablespoons of flour over the leeks and stir constantly for 1 minute to cook the flour. Slowly pour in the chicken broth and heavy cream, stirring constantly to avoid lumps. Bring the mixture to a simmer and cook for 3 to 4 minutes until it thickens to a gravy-like consistency.
Stir in the cooked chicken, thyme, and black pepper. Remove the skillet from the heat and let the filling cool slightly.
Preheat your oven to 375°F. On a lightly floured surface, roll out one disk of dough into a 12-inch circle and carefully transfer it to a 9-inch pie dish. Gently press the dough into the bottom and sides of the dish. Pour the slightly cooled chicken and leek filling into the crust.
Roll out the second disk of dough into an 11-inch circle. Carefully place it over the filling. Trim any excess dough, then crimp the edges together with your fingers or a fork to seal. Cut 4 small slits in the top crust to allow steam to escape. Brush the entire top crust lightly with the egg wash.
Bake the pie on a baking sheet for 45 to 50 minutes, or until the crust is deep golden brown and the filling is bubbling. Let the pie rest on a wire rack for at least 15 minutes before slicing and serving.