Combine flour, salt, and sugar in a large bowl. Add cold butter cubes and press into the flour with your fingertips until the mixture looks like rough crumbs with some pea-sized butter pieces. Add ice water one tablespoon at a time, mixing with a fork, until the dough just comes together when pressed.

Divide dough into two flat disks, wrap each in plastic wrap, and refrigerate for at least 30 minutes.

Melt 4 tbsp butter in a large pot over medium heat. Add onion, celery, and carrots. Cook 6-8 minutes until softened. Add garlic and cook 1 minute.

Sprinkle flour over the vegetables and stir constantly for 2 full minutes until the raw flour smell is gone.

Pour in chicken broth in a slow stream while whisking. Add cream. Simmer 3-4 minutes, stirring, until the filling thickens to coat the back of a spoon. Add thyme, rosemary, salt, and pepper.

Stir in chicken and frozen peas. Taste and adjust seasoning. Cool for at least 10-15 minutes.

Preheat oven to 200°C (400°F). Roll one dough disk into a 30cm circle and line a 23cm deep-dish pie plate, letting excess hang over the edge.

Pour cooled filling into the crust. Roll the second disk into a 30cm circle and lay over the filling. Fold and crimp the edges to seal. Cut 4-5 steam vents. Brush the full surface with egg wash.

Place pie on a rimmed baking sheet. Bake 40-45 minutes until the crust is deep golden brown and the filling is bubbling through the vents. Cover crust edges with foil if they darken early.

Rest on a wire rack for 10 minutes before cutting.