Heat olive oil in a large skillet with a lid over medium-high heat.
Add sliced kielbasa or smoked sausage in a single layer. Cook for 4 to 5 minutes, turning once, until browned. Transfer the sausage to a plate.
Add diced potatoes to the same skillet. Season with smoked paprika, caraway seeds if using, black pepper, and salt. Stir well.
Add chicken broth or water. Cover the skillet and cook for 8 to 10 minutes, stirring once halfway through, until the potatoes begin to soften.
Remove the lid and cook potatoes for 5 to 7 more minutes, stirring occasionally, until golden and fork-tender.
Push the potatoes to one side of the skillet. Add sliced onion and cook for 2 to 3 minutes until softened.
Add garlic and cook for 30 seconds, stirring constantly.
Add drained sauerkraut, Dijon mustard, diced apple if using, and apple cider vinegar if using. Stir everything together.
Return the browned sausage to the skillet. Reduce the heat to medium-low, cover, and cook for 5 to 7 minutes until hot and flavorful.
Taste and adjust seasoning. Add more mustard, black pepper, or a small splash of broth if needed.
Garnish with chopped parsley or dill. Serve hot with mustard and bread if desired.