Season the ground beef all over with salt and pepper. Heat a large stock pot over medium heat and add beef. Brown the beef, using a meat chopped or wooden spoon to break it into small crumbles.
Once browned, remove the beef with a slotted spoon and transfer it to a plate.
Add the olive oil to the same skillet. Add in the onions, carrots, celery and garlic with a big pinch of salt and pepper. Add the oregano. Stir. Cook, stirring often, until the veggies soften, about 5 to 6 minutes.
Add in the potatoes and 4 cups of chicken stock. Return the beef to the pot. Bring the mixture to a boil, then reduce to a simmer and cover. Cook until the potatoes are tender, about 10 to 15 minutes.
Heat the butter in a separate skillet. Once melted, whisk in the flour to create a roux. Whisk and cook for 2 to 3 minutes, until the color is golden and the mixture is fragrant.
Use a spatula to scrape the roux right into the soup pot. Bring the mixture back to a boil. Boil for 1 to 2 minutes. Add in the milk and cook for another 1 to 2 minutes.
Reduce the heat to low and stir in the cheese. Stir until it is all melted. At this point, you can add in the additional cup of chicken stock or not. Sometimes I like to add it before I pack the soup up for the fridge since it thickens more as it sits. Taste and season the soup with more salt and pepper if necessary. Stir in the sour cream. Once melted, serve! Top with fresh chives to serve, or chips if desired.