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

Sauté the vegetables: In a large skillet, melt the butter over medium heat. Add onion, carrots, and celery with a pinch of salt. Cook until softened, about 6–8 minutes. Stir in the garlic and cook 30 seconds more.

Make the roux: Sprinkle the flour over the veggies and stir to coat. Cook for 1–2 minutes to remove the raw flour taste.

Build the sauce: Slowly whisk in the chicken broth, scraping up any bits from the pan. Add the milk or half-and-half. Simmer, stirring often, until thick and creamy, about 3–5 minutes.

Season: Add thyme, parsley, black pepper, and salt to taste. The sauce should be well-seasoned, since it will mellow once mixed with chicken and veggies.

Combine the filling: Stir in the cooked chicken, peas, and any optional add-ins like corn or parboiled potatoes. If the mixture seems too thick, splash in a little more broth or milk. If too thin, simmer another minute.

Assemble: Transfer the mixture to your prepared baking dish and smooth the top.

Add the topping: Lay puff pastry or pie crust over the filling, tucking edges to fit. Cut a few slits for steam. If using biscuit dough, arrange biscuits evenly over the top. Brush with egg wash for a glossy finish, if you like.

Bake: Place on the center rack and bake 25–35 minutes, until the topping is golden and the filling is bubbling around the edges. If using biscuit dough, ensure the bottoms of the biscuits are cooked through.

Rest and serve: Let the casserole rest for 10 minutes before serving. This helps the filling set and makes clean slices easier.