Preheat the oven to 375°F (190°C).
Melt 4 tablespoons of butter in a large skillet over medium-high heat.
Add onion, carrots, celery, thyme, salt, and pepper. Sauté for 5–7 minutes until vegetables soften.
Stir in garlic and flour, cooking for 30–60 seconds until fragrant.
Slowly pour in chicken broth and milk, scraping the pan bottom. Bring to a boil.
Reduce heat and simmer 4–5 minutes until the sauce thickens.
Stir in chicken and frozen peas. Taste and adjust seasoning.
Transfer the mixture to a greased 9×13-inch baking dish and spread evenly.
Arrange biscuit dough pieces in a single layer over the filling.
Bake for 20–30 minutes until biscuits are golden and the filling is bubbling.
Melt remaining 2 tablespoons butter and brush over biscuits before serving.