Sauté onion, celery and garlic in a large saucepan or shallow kettle in a little of the butter over medium heat a minute or two until sizzling hot and bubbling thoroughly. Add the oysters and continue to sauté them until the edges curl up, or they are warmed through. Add the rest of the butter and the remaining dairy products, salt, spices and parsley and/or chives. Heat UNCOVERED slowly on very low heat.
The last thing you want to do is bring this anywhere near a boil because boiling curdles the milk (it may take 45 minutes to an hour - or typically longer - to bring it to a nice steamy temperature on low heat; this is definitely not a "30-minute" meal, but it is so savory-good and worth the wait)!