Preheat your oven to 400°F (200°C) and position the rack in the center.

In a large bowl, combine 2 cups of all-purpose flour and 1 cup of cold butter, cutting until coarse breadcrumbs.

Gradually add 4–6 tablespoons of ice-cold water and stir until the dough clumps, then shape into disks and refrigerate for 30 minutes.

On a floured surface, roll one disk into an 11-inch circle and fit into a 9-inch pie dish, leaving a 1-inch overhang.

Season chicken breasts with salt and pepper, then poach in 2 cups of chicken broth for 8 minutes. Remove, cool slightly, and shred.

Melt 4 tablespoons of butter in a skillet over medium heat, then add onions, carrots, and celery, cooking until softened, about 5 minutes.

Stir in minced garlic and sprinkle in 1/4 cup flour, cooking until pale golden.

Whisk in the reserved chicken broth and 1/2 cup heavy cream, bringing to a gentle simmer for about 3–5 minutes.

Fold shredded chicken and frozen peas into the sauce, season with salt and pepper.

Spoon the filling into the prepared crust, creating a slight mound in the center.

Roll second dough disk over the filling, trim excess, crimp edges, and cut steam vents on top.

Brush the top crust with beaten egg for a golden finish.

Bake at 400°F for 35–40 minutes until crust is golden and filling is bubbly.

Let the pie rest on a wire rack for at least 10 minutes before slicing.