In a large skillet, heat the vegetable oil over medium heat and add the potato. Saute the potato until the pieces start to turn translucent and brown on the edges, about 5 minutes.
Add the onion and kielbasa. Toss until kielbasa browns and onion softens, about 5 minutes.
Stir in the rinsed and drained sauerkraut, apple juice and brown sugar, mixing until combined. Reduce to a low and simmer for 10-12 minutes.
Season with the salt and freshly ground black pepper.
Serve immediately.
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Notes
I use apple juice for the liquid, but you can also use chicken stock, plain water or even dry white wine.