Step 1: Prepare the Pie Crust : In a large bowl, whisk together flour and salt. Add cold butter cubes and cut them into the flour using a pastry cutter or fingertips until mixture resembles coarse crumbs. Gradually add ice water, 1 tbsp at a time, mixing just until dough comes together. Divide dough into two discs, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Step 2: Make the Chicken Filling : In a large skillet, melt butter with olive oil over medium heat. Add onions, garlic, carrots, and celery. Sauté for 5–7 minutes, until softened. Sprinkle in flour and cook for 1 minute, stirring constantly (this helps thicken the sauce). Gradually whisk in chicken broth, followed by milk or cream. Stir until smooth and thickened (about 3–4 minutes). Add shredded chicken, peas, salt, pepper, thyme, and nutmeg. Mix well, then remove from heat and let the filling cool slightly.

Step 3: Assemble the Pie : Preheat oven to 400°F (200°C). On a floured surface, roll out one disc of dough to fit a 9-inch pie dish. Place the crust into the dish, pressing gently to fit. Spoon the cooled chicken filling into the crust evenly. Roll out the second disc of dough and lay it over the top. Trim excess edges and crimp to seal. Cut a few small slits in the center to allow steam to escape. Brush top crust with beaten egg for a glossy golden finish.

Step 4: Bake the Pie : Bake for 35–40 minutes, until the crust is golden brown and crisp. If edges brown too quickly, cover them with foil. Let the pie rest for 10–15 minutes before slicing to allow filling to set.