1. In a six-quart pot or Dutch oven, over medium-low heat, heat two tablespoons olive oil. Add flanken and brown on both sides, about eight minutes total.
2. Remove flanken from pan and pour off excess fat if there is too much. Add remaining tablespoon olive oil and garlic. Cover and cook until garlic is tender but not browned, about two minutes.
3. Add onion and sauté until translucent, about four minutes. Add three cups water, carrots, crushed tomatoes, tomato paste, ketchup, brown sugar and bay leaf. Simmer until carrots are tender, about 15 minutes. Discard bay leaf.
4. Using an immersion blender, or working in batches with a stand blender, process mixture until it is coarse, not fully puréed.
5. Return to pot and add lemon juice, cabbage, and three cups water. Add flanken back to pot. Place over medium-high heat and simmer until cabbage and flanken are cooked, about an hour and 15 minutes to an hour and a half.
6. Add three to six cups water to thin to desired consistency. Ten minutes before serving, season with kosher salt and pepper to taste. Serve warm.