Preheat the oven to 400°F (200°C). Set a rack in the center.
Keep your pie dough chilled until you’re ready to top the pie—cold dough equals a flakier crust.
Sauté the vegetables. In a large skillet, heat butter and olive oil over medium heat. Add onion, carrots, and celery with a pinch of salt. Cook 6–8 minutes, stirring, until softened.
Add garlic and cook 30 seconds.
Make the roux. Sprinkle flour over the vegetables and stir for 1–2 minutes. It should look a bit pasty. This step thickens the sauce later.
Add liquids and season. Slowly whisk in chicken broth, then milk.
Add thyme, bay leaf (if using), 1/2 teaspoon salt, and 1/4 teaspoon pepper. Simmer 4–6 minutes, stirring, until thick and creamy. Remove bay leaf.
Stir in chicken and peas. Fold in the cooked chicken and frozen peas.
Add parsley and lemon juice if using. Taste and adjust salt and pepper. The filling should be well-seasoned and spoonable, not runny.
Assemble the pie. Transfer filling to a 9-inch pie dish.
Drape the chilled pie crust over the top. Trim any excess, then crimp the edges to seal. Cut 4–5 small slits in the center to vent steam.
Brush and bake. Brush the crust with egg wash for shine.
Place the dish on a baking sheet to catch drips. Bake 28–35 minutes, until the crust is deep golden and the filling is bubbling at the edges.
Rest before serving. Let the potpie rest 10–15 minutes to set. This keeps slices neat and prevents the filling from spilling out.