Melt half the butter in a large pan or wok and fry the ground beef over high heat until browned. Use a slotted spoon to remove the beef and set aside, leaving the fat and juices in the pan.
Add the remaining butter to the pan and add the onion, celery, and carrot. Fry over moderate heat until the vegetables soften and start to caramelize.
Return the beef to the pan and add the dried herbs and the chicken broth, reserving a few tablespoons of the broth in a small bowl. Bring the mixture to a light simmer for approximately 5 minutes until the soup reduces slightly.
In the meantime, dissolve the arrowroot powder in the little bowl of stock set aside. Add this to the pan, stirring well to combine.
Add the grated cheese (reserving a little for garnish) and stir well until completely melted and the soup reduces further and thickens.
Divide the mixture between two warm bowls and garnish with a dollop of sour cream, the remaining grated cheese, and any chopped herbs of your choice.