Heat oil or butter in a large pot over medium heat; sauté onion, carrot, and celery until softened, about 6 minutes.
Add garlic and cook 30 seconds, then add chicken and cook until lightly browned.
Stir in wild rice, broth, thyme, and bay leaf; bring to a boil, then reduce to a simmer and cover.
Simmer 35–45 minutes until rice is tender and chicken is cooked through.
If using milk + flour, whisk together and stir in; or add half-and-half and heat gently until slightly thickened. Season with salt, pepper, and parsley.