Step 1: Heat the olive oil in a large, deep skillet or Dutch oven over medium-high heat. Once shimmering, add the ground beef to the skillet. Break it up with a spoon and cook, stirring occasionally, until it's completely browned and no pink remains. Drain any excess grease from the pan.
Step 2: Reduce the heat to medium. Add the chopped onion to the skillet with the browned ground beef. Sauté for 5-7 minutes, stirring occasionally, until the onion has softened and turned translucent.
Step 3: Add the minced garlic to the skillet and cook for another minute until fragrant, being careful not to burn it.
Step 4: Stir in the chopped cabbage. Continue to cook for 5-10 minutes, stirring occasionally, until the cabbage has softened and reduced in volume.
Step 5: Pour in the undrained diced tomatoes, tomato sauce, beef broth, apple cider vinegar, brown sugar (if using), smoked paprika, and dried thyme. Stir everything together until well combined.
Step 6: Bring the mixture to a gentle simmer. Once simmering, reduce the heat to low, cover the skillet, and let it cook for 15-20 minutes, stirring occasionally, until the cabbage is wonderfully tender.
Step 7: Season generously with salt and freshly ground black pepper to taste. Adjust as needed.
Step 8: Remove from heat and let it rest for a few minutes before serving. Garnish with fresh chopped parsley, if desired.