In a large skillet or Dutch oven, melt the butter over medium heat. Add the onion and celery; cook until soft, about 5 minutes. Add the garlic and cook for 1 more minute, stirring constantly. Whisk in the flour; cook, whisking constantly, for 1 minute. Gradually whisk in the chicken broth and milk until smooth. Add the salt, pepper, and thyme. Cook, whisking frequently, until thickened, about 4-5 minutes.