Warm a large skillet over medium-high heat. Add 1 tablespoon oil and the ground beef.
Cook, breaking it up with a spatula, until browned and cooked through, 5–7 minutes. Season with salt and pepper. Transfer to a plate and drain excess grease if needed.
In the same skillet, add the remaining oil and the onion.
Sauté 2–3 minutes until softened and fragrant. Add garlic and cook 30 seconds more.
Add the sliced cabbage. Season with salt, pepper, smoked paprika, oregano, and red pepper flakes.
Toss well to coat and cook 5–8 minutes, stirring occasionally, until the cabbage softens and starts to wilt.
Pour in the broth. Scrape up any browned bits from the bottom of the pan. Let it simmer 2–3 minutes to reduce slightly.
Stir in the cooked beef and any juices.
Add butter and heavy cream. Toss everything together and cook 2–3 minutes, until the sauce lightly coats the cabbage.
Reduce heat to low and sprinkle cheese evenly over the top. Cover the skillet for 1–2 minutes, just until the cheese melts.
Taste and adjust seasoning.
Finish with chopped parsley or green onions. Squeeze a little lemon over the top if you like. Serve hot.