Step 1: Heat olive oil in a large, heavy-bottomed pot or Dutch oven over medium-high heat. Add the sliced smoked sausage and cook for 5-7 minutes, turning occasionally, until browned and slightly crispy. Remove the sausage with a slotted spoon and set aside, leaving the drippings in the pot.
Step 2: Add the sliced onion to the pot with the sausage drippings. Sauté for 5-7 minutes, stirring occasionally, until softened and translucent. Scrape up any browned bits from the bottom of the pot.
Step 3: Add the minced garlic to the pot and cook for an additional 1 minute until fragrant. Be careful not to burn the garlic.
Step 4: Add the shredded cabbage to the pot. It will look like a lot, but it will cook down. Stir well to combine with the onions, garlic, and drippings. Cook for 5-10 minutes, stirring occasionally, until the cabbage begins to wilt.
Step 5: Pour in the chicken broth and apple cider vinegar. Return the cooked smoked sausage to the pot. Season with caraway seeds (if using), black pepper, and a pinch of salt. Stir everything together.
Step 6: Bring the mixture to a simmer, then reduce the heat to low. Cover the pot and let it braise for 20-30 minutes, or until the cabbage is tender to your desired consistency. Stir occasionally during cooking.
Step 7: Taste the braised cabbage and adjust seasonings as needed. Add more salt, pepper, or a splash more apple cider vinegar for brightness if desired.
Step 8: Serve hot, garnished with fresh chopped parsley or dill, if using. Enjoy your hearty braised cabbage with smoked sausage!