Prep the oven and dish: Preheat your oven to 400°F (200°C).
Grease a 9x13-inch baking dish and set aside.
Sauté the aromatics: In a large skillet or pot, heat the butter and olive oil over medium heat. Add onion, carrots, and celery. Cook 6–8 minutes until softened.
Stir in garlic and cook 30 seconds.
Build the sauce: Sprinkle the flour over the vegetables and stir to coat. Cook 1–2 minutes until the raw flour smell fades. Slowly whisk in the chicken broth, scraping up any bits.
Add milk or half-and-half and stir until smooth and slightly thickened.
Season and add veggies: Stir in thyme, parsley, sage, paprika, salt, and pepper. Add the frozen peas and corn. Simmer 3–4 minutes until the sauce is creamy and coats a spoon.
Add the chicken: Fold in the cooked chicken.
Taste and adjust seasoning. Transfer the filling to the prepared baking dish and spread evenly.
Make the biscuit topping: In a bowl, whisk flour, baking powder, baking soda, salt, and pepper. Cut in the cold butter with a pastry cutter or your fingers until the mixture looks like coarse crumbs.
Stir in buttermilk and parsley just until combined. Do not overmix.
Top the casserole: Drop spoonfuls of the biscuit dough over the filling, leaving small gaps for steam. For a shinier top, brush with egg wash.
Bake: Bake 25–30 minutes, until the topping is golden and the filling is bubbling around the edges.
If browning too fast, tent loosely with foil.
Rest and serve: Let the casserole rest 10 minutes so the sauce thickens slightly. Serve warm.