Heat oil in a large saucepan over medium heat.
Add celery, carrot, onion, and thyme and saute for about 5 minutes until the vegetables begin to soften.
Sprinkle over the flour and stir for about 2 minutes.
Gradually whisk in broth, then milk.
Add the potato and bring the soup to the boil.
Reduce heat and simmer soup for about 20 minutes until the potato is tender.
Add cheese 1/3 of a cup at a time, stirring until melted and smooth after each addition then mix in the ham.
Season soup to taste with hot pepper sauce, salt, and pepper.
Sprinkle with parsley and serve