Preheat the oven to 375°F (190°C). Lightly grease a 9 × 13-inch baking dish.
Melt the butter in a large skillet over medium heat. Add the diced onion and cook for 3–4 minutes until softened.
Add the minced garlic and cook for about 30 seconds, stirring frequently.
Sprinkle the flour over the onion and garlic. Stir for about one minute.
Slowly whisk in the chicken broth, followed by the milk. Cook for 3–5 minutes, stirring until the sauce thickens.
Stir in the dried thyme, black pepper, and salt to taste.
Add the cooked chicken and frozen mixed vegetables. Stir until evenly coated and heated through.
Transfer the filling to the prepared baking dish and spread it into an even layer.
Arrange the refrigerated biscuits over the filling, leaving small gaps between them.
Bake for approximately 25–35 minutes, or until the biscuits are golden and fully cooked and the filling is bubbling. Follow the biscuit package directions if they differ.
Check that the chicken filling reaches 165°F (74°C) in the center.
Let the casserole rest for 5–10 minutes before serving.