Make a roux from 4 tablespoons of the olive oil and flour. Set aside when thick.
Brown the beef, carrots, and onion in the remaining two tablespoons of olive oil. Allow to simmer for about ten minutes.
Add potatoes, thyme, broth, and bloody mary mix. Bring to a boil, then add roux, stirring well to combine.
Add canned tomatoes, brown sugar, and cabbage (one-quarter head at a time). Stir gently and cover. (Yes, this will seem like a lot of cabbage. It cooks down really well. Don't worry!)
Turn heat to medium-low and allow to simmer for about 30 minutes, stirring occasionally. Just before serving, taste. Add salt & pepper to your taste.