Preheat and prep: Heat oven to 400°F (200°C). Grease a 9x13-inch baking dish or similar casserole.

Sauté aromatics: In a large skillet, warm olive oil and 2 tablespoons butter over medium heat. Add onion, carrots, and celery. Cook 6–8 minutes, stirring, until softened. Add garlic and cook 30 seconds.

Make a roux: Add remaining 2 tablespoons butter. When melted, sprinkle in the flour. Stir constantly for 1–2 minutes until the flour smells toasty and clings to the veggies.

Build the sauce: Slowly pour in the chicken broth, whisking or stirring to avoid lumps. Add milk or half-and-half, thyme, bay leaf, paprika or poultry seasoning, and a big pinch of salt and pepper. Simmer 3–5 minutes until thick and creamy.

Add chicken and veggies: Stir in the cooked chicken, peas, and corn. Let it bubble for 1–2 minutes. Taste and adjust seasoning. Remove the bay leaf. The mixture should be spoonable but not soupy.

Assemble: Spoon the filling into the greased dish. If using pie crust, lay the crust on top, trimming excess and cutting a few slits to vent. Brush with egg wash and sprinkle with a pinch of salt. If using biscuits, place them evenly on top, leaving a little space between each.

Bake: Place on a rimmed sheet to catch drips. Bake 22–28 minutes for pie crust, or 18–24 minutes for biscuits, until the top is golden and the filling is bubbling around the edges.

Rest and serve: Let the casserole rest 10 minutes so the sauce sets slightly. Sprinkle with fresh parsley. Serve warm.