Step 1: Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the sliced smoked sausage and cook, stirring occasionally, until browned and slightly crispy, about 5-7 minutes. Remove the sausage with a slotted spoon and set aside, leaving the rendered fat in the pot.
Step 2: Add the chopped onion, green bell pepper, and celery to the pot. Cook, stirring occasionally, until softened, about 5-7 minutes. If the pot seems dry, add a splash more olive oil.
Step 3: Stir in the minced garlic, dried thyme, smoked paprika, cayenne pepper, salt, and black pepper. Cook for another minute until fragrant, stirring constantly to prevent burning.
Step 4: Add the chopped cabbage to the pot. It will seem like a lot, but it will cook down. Stir well to combine with the aromatics and spices. Cook for 5-7 minutes, stirring occasionally, until the cabbage begins to wilt.
Step 5: Pour in the diced tomatoes (undrained) and chicken broth. Add the apple cider vinegar and brown sugar (if using). Stir everything together, scraping up any browned bits from the bottom of the pot. Return the cooked sausage to the pot.
Step 6: Bring the mixture to a simmer, then reduce the heat to low, cover the pot, and let it braise for 30-40 minutes, or until the cabbage is very tender. Stir occasionally to ensure even cooking and prevent sticking. Taste and adjust seasonings as needed.
Step 7: Once the Dirty Cajun Braised Cabbage is tender and the flavors have melded, remove it from the heat. Garnish with fresh chopped parsley, if desired, and serve hot.