Slice the kielbasa into bite-sized pieces.
Heat olive oil in a large skillet and add the sausages.
Cook on medium-low heat until brown, which should take 8-10 minutes.
Flip the pieces, so that both sides are evenly cooked.
Remove the sausages from the skillet.
Using the same skillet, melt the butter and add the onions.
Sauté until brown.
Once the onions turn brown, add garlic and cook for a minute.
Add the brown sugar, cooking until the sugar dissolves.
Add the sauerkraut to the skillet, and then add the sausages on top of the sauerkraut.
Stir well and cook for five minutes until the sauerkraut is warm.
Season with salt and pepper before serving.