In a large bowl, whisk together all-purpose flour, sugar, and salt until well combined. Cut in cold butter using a pastry cutter or your fingers, mixing until the mixture resembles coarse crumbs. Add buttermilk and ice water gradually, stirring just until combined. Form the dough into a ball, wrap in plastic, and chill in the refrigerator for at least 1 hour to ensure a flaky crust.

In a large skillet over medium heat, melt butter. Add diced onion, carrots, celery, and minced garlic, sautéing for 5–7 minutes until the vegetables are tender and fragrant. Stir frequently to prevent burning.

Pour in chicken broth and heavy cream, stirring continuously until the mixture begins to thicken and bubble, about 3–5 minutes. Keep on low heat, allowing it to achieve a creamy consistency while stirring occasionally. Fold in shredded cooked chicken and frozen peas, cooking just until heated through. Remove from heat and set aside to cool slightly.

Preheat your oven to 400°F (200°C). As it warms, take your chilled dough from the fridge and divide it into two equal portions. On a lightly floured surface, roll out one half of the dough into a large circle, about 1/8 inch thick. Carefully transfer this dough into a 9-inch pie pan.

Pour the chicken filling into the prepared crust, ensuring it is evenly distributed. Roll out the second half of the dough to form the top crust. Place it over the filled pie, crimping the edges to seal and cut several slits in the top crust.

Brush the top crust with egg wash for a beautiful golden finish. Bake in the preheated oven for about 45 minutes, or until the crust is golden brown and the filling is bubbling.

Once baked, remove the Chicken Pot Pie from the oven and let it cool for 10 minutes before serving.