Brown the Chorizo: Heat a large skillet over medium heat. Add the ground chorizo and cook until nicely browned, breaking it up as it cooks. Once browned, remove the chorizo with a slotted spoon, leaving any rendered fat in the skillet. Set the cooked chorizo aside.
Sauté Aromatics: To the same skillet, add the diced poblano pepper, diced onion, butter, and half of the salt. Sauté, stirring occasionally, until the vegetables are nearly tender, about 5-7 minutes.
Build Flavor: Push the sautéed vegetables to the sides of the skillet to create a space in the center. Add the minced garlic to the center and sprinkle the smoked paprika over the peppers and onions. Cook for about 20 seconds, stirring the garlic *just* until fragrant, being careful not to let it burn.
Add Liquids & Cabbage: Stir in the undrained diced tomatoes, tomato paste, and chicken broth. Mix well. Then, add the chopped cabbage and the remaining salt. Cover the skillet or cook, stirring often, until the cabbage softens to your liking, about 10-15 minutes.
Combine & Finish: Return the cooked chorizo to the skillet and stir everything together, heating through for a couple of minutes. Finally, stir in the fresh chopped cilantro and lime juice. Serve hot and enjoy!