Preheat the oven: Set your oven to 400°F (200°C) so it’s ready once the filling is done.
Cook the vegetables: In a large skillet, heat the olive oil or butter over medium heat. Add the onion, carrots, and celery. Cook until softened, about 5–7 minutes.
Add garlic and peas: Stir in the garlic and frozen peas, cooking for 1–2 minutes more until fragrant.
Make the roux: Sprinkle the flour over the vegetables and stir well to coat everything. Let it cook for 1–2 minutes to remove the raw flour taste.
Add broth and milk: Slowly pour in the chicken broth and milk, stirring constantly until the mixture starts to thicken and bubble.
Stir in the chicken: Add the shredded chicken to the creamy mixture and season with salt, pepper, and thyme. Let it simmer for another 3–4 minutes.
Transfer to casserole dish: Pour the filling into a greased 9x13 baking dish and spread it out evenly.
Top with biscuit dough: Drop spoonfuls of biscuit dough over the top, spacing them evenly so they have room to rise.
Bake the casserole: Place the dish in the oven and bake uncovered for 20–25 minutes, or until the biscuits are golden brown and cooked through.
Let it rest: Remove from the oven and let the casserole sit for 5–10 minutes before serving.
Garnish and enjoy: Sprinkle with chopped fresh parsley or thyme if you like, and serve warm.