Preheat the oven: Heat to 400°F (200°C). Lightly butter a 9x13-inch baking dish or similar casserole.
Sauté the aromatics: In a large skillet or Dutch oven over medium heat, melt the butter with the olive oil.
Add onion, carrots, and celery. Cook, stirring, until softened, about 6–8 minutes. Stir in the garlic and cook 30 seconds until fragrant.
Build the roux: Sprinkle flour over the vegetables and stir to coat.
Cook 1–2 minutes, stirring constantly, until the flour smells toasty but not burned.
Make the sauce: Gradually whisk in the chicken broth, then the milk. Add thyme, bay leaf, Dijon (if using), salt, and pepper. Bring to a gentle simmer and cook 3–5 minutes, stirring, until thick and creamy.
Remove and discard the bay leaf.
Add the fillings: Stir in the chicken, peas, and corn (if using). Taste and adjust seasoning with more salt and pepper as needed. If using parsley, stir it in now.
Transfer to the dish: Pour the filling into the prepared casserole dish and smooth the top.
Top with pastry: Lay the puff pastry (or pie crust) over the filling.
Trim or tuck the edges to fit. Use a sharp knife to cut 4–6 small slits for steam to escape. Brush the top lightly with egg wash.
Bake: Place the casserole on a baking sheet to catch any drips.
Bake 25–35 minutes, until the pastry is deeply golden and the filling is bubbling at the edges.
Rest and serve: Let it rest 10 minutes before serving so the filling sets slightly. Spoon into bowls and enjoy.