Heat a large cast-iron skillet over medium-high heat. Add olive oil if needed.
Add ground beef and cook until deeply browned, breaking it up as it cooks.
Add cabbage to the skillet and spread it out in an even layer.
Cook without stirring for 3–4 minutes to allow the cabbage to char.
Stir and continue cooking for 4–5 minutes until cabbage is tender with browned edges.
Add garlic, smoked paprika, cumin, salt, and black pepper.
Cook for 30 seconds until fragrant, stirring to combine.
Remove from heat and squeeze fresh lime juice over the skillet before serving.