Begin by preparing your ingredients. With a sharp chef's knife, slice the onion thinly, mince the garlic, and cut the sausage into half-inch pieces. Peel, core, and dice the apple into small cubes. Set these aside as you heat the skillet.
Warm the olive oil in a large skillet over medium heat. Once the oil shimmers, introduce the sliced onions, stirring them occasionally. You're looking for them to soften and become translucent, a process that should take about 5 minutes.
Add the minced garlic to the skillet, stirring it with the onions for about a minute until its aroma blooms, a hint that it's time to add the sausage.
Slide the sausage pieces into the skillet. Let them brown gently, stirring occasionally, for about 8 minutes. The sausage should have a nice, light caramelization on the outside.
Now, it's time to incorporate the sauerkraut. After draining and rinsing it, add the sauerkraut to the skillet, along with the diced apple. Stir to combine.
Pour in the chicken or vegetable broth, ensuring it deglazes the skillet, lifting any browned bits from the bottom. This will infuse your dish with deep flavors.
Sprinkle the brown sugar over the mixture, and if you're using them, scatter the caraway seeds in as well. These will add a subtle, earthy depth to the dish.
Season with salt and pepper, but do so judiciously. Remember, the sauerkraut and sausage already carry their own saltiness.
Cover the skillet with a lid, reduce the heat to low, and let the concoction simmer for about 20 minutes. You want the flavors to meld together, for the sauerkraut to soften further, and for the apple to infuse its sweetness throughout.
After the simmering time has elapsed, remove the lid and give the mixture a good stir. Taste it, adjusting the seasoning if necessary.
To serve, spoon the sauerkraut and sausage skillet onto plates. Garnish each serving with a sprinkle of fresh parsley, adding a pop of color and a fresh flavor contrast.