Heat oil or butter in a large pot over medium heat
Add chopped onion, carrot, and celery
Cook until softened
Add minced garlic and cook briefly
Add diced chicken and cook until no longer pink
Season with salt, black pepper, and dried thyme
Sprinkle in flour and stir to coat
Slowly pour in chicken broth while stirring
Add diced potatoes if desired
Simmer until vegetables are tender
Stir in milk, cream, or half-and-half
Add corn, peas, or mushrooms if desired
Cook gently until heated through
Adjust seasoning to taste
Serve hot
Garnish with parsley if desired