Set a large skillet over medium heat. Add the butter and olive oil and cook until the butter melts.
Add the onion and garlic and cook, stirring frequently, until the onion is soft, about 5 minutes.
Stir in the flour and let it cook, stirring frequently, for 3 minutes. The mixture will be a light golden brown color.
Slowly whisk in the broth and milk. Bring to a simmer and continue to cook until thickened—it should be thick enough that it coats your spoon, about 5 to 8 minutes. Stir in the salt and pepper.
Stir in the chicken along with the thawed mixed vegetables. Simmer until the chicken and vegetables are heated through, about 5 minutes.