Over medium-high heat, brown ground beef, breaking up large pieces, until no longer pink, about 5 minutes. Add celery and onion; cook, stirring, until the vegetables are softened.
Meanwhile, in a medium saucepan, melt butter over medium heat. Whisk in flour. Slowly add milk, whisking or stirring constantly, and cook until thickened, about 2 minutes.
Whisk in tomato paste, ketchup and mustard. Remove from heat. Slowly stir the milk mixture into the soup; bring to a boil. Reduce heat to low and gradually stir in cheese.
When cheese is completely incorporated into the soup, remove from heat and stir in chives. Don’t boil the soup after the cheese has been added. Serve topped with tomatoes, onion and/or pickles.