Preheat your oven to 375°F (190°C). Lightly grease a 9×13-inch baking dish or similar casserole pan with a bit of olive oil or butter.
In a large skillet, heat the butter and olive oil over medium heat. Add the sliced onion and cook, stirring often, for 8–10 minutes, until soft and lightly golden. Stir in the minced garlic and cook for another 30–60 seconds, just until fragrant.
Push the onions to one side of the skillet and add the sliced smoked sausage or kielbasa. Cook for 4–5 minutes, stirring occasionally, until the sausage gets some color on the edges. You are just searing for flavor, not fully cooking through.
Turn off the heat and add the drained sauerkraut and sliced apple to the skillet. Toss gently to combine with the onions and sausage. The mixture will look rustic and loosely combined.
Scatter the halved baby potatoes across the bottom of the prepared baking dish, trying to place most of them cut-side down for better browning and flavor.
In a small bowl or measuring cup, whisk together the chicken broth, whole-grain mustard, caraway seeds, smoked paprika, and black pepper. Taste and adjust, keeping in mind that the sausage and sauerkraut are already salty; do not add extra salt yet.
Spoon the sauerkraut–sausage–onion mixture evenly over the potatoes in the baking dish. Pour the seasoned broth mixture evenly over the top so everything gets a bit of moisture.
Cover the baking dish tightly with foil and bake for 35 minutes. This covered time helps the potatoes steam and the flavors meld.
Remove the foil, gently stir once or twice to bring some potatoes toward the top, and continue baking uncovered for 20–25 minutes, until the potatoes are tender, the sausage edges are lightly browned, and the sauerkraut on top is starting to caramelize.
Remove the casserole from the oven and let it rest for 5–10 minutes. Taste and add salt if needed. Garnish with chopped fresh parsley before serving, if desired.