Cut cold butter into the flour, salt, and thyme until most pieces are pea-sized. Add 5 tablespoons ice water and toss, adding more 1 tablespoon at a time until a squeezed handful holds together. Press into a disk, wrap, and refrigerate for 45 minutes.
Season chicken with 1 teaspoon salt and ½ teaspoon pepper. Working in two batches, brown in 1 tablespoon oil per batch over medium-high heat for 4–5 minutes, turning once. Transfer to a plate; centers can remain underdone.
Reduce heat to medium and melt butter in the skillet. Cook leek, carrots, celery, and remaining salt for 8–10 minutes until softened. Add garlic and sage for 30 seconds, then stir in flour for 2 minutes until lightly toasted.
Gradually stir in broth in three additions, scraping the base, then stir in milk and remaining pepper. Simmer for 5–7 minutes until a spoon drawn through the skillet leaves a slowly closing trail. Return chicken and plate juices, simmer 5 minutes until chicken reaches 165°F (74°C), then fold in peas, parsley, lemon zest, and lemon juice off the heat.
Transfer filling to a 9-inch deep pie dish and cool uncovered for 20–30 minutes. Preheat oven to 425°F (220°C). Roll dough into an 11-inch circle, lay over the dish, trim to a 2 cm overhang, fold under, press against the rim, and cut 4 vents. Brush with egg wash and place on a rimmed sheet pan.
Bake at 425°F for 15 minutes, then reduce to 375°F (190°C) and bake for 25–30 more minutes until the crust is deep gold and filling bubbles through a vent. Rest for 20 minutes before portioning.