Thinly slice ½ large red onion if using. Drain 1 pound sauerkraut. Cut about 14 ounces kielbasa sausage crosswise into ½-inch-thick rounds.

Heat 1 tablespoon unsalted butter and 1 tablespoon vegetable oil in a large skillet over medium heat until the butter is melted and the oil is shimmering. Add the kielbasa in an even layer. Cook until browned on the bottom, about 3 minutes. Flip and cook until browned on the second side, about 3 minutes more. Transfer to a bowl or plate using tongs or a slotted spoon, leaving the fat in the skillet.

Add the sauerkraut, red onion if using, 2 tablespoons packed light brown sugar, and 1 tablespoon spicy brown, whole-grain, or Dijon mustard if using to the skillet. Cook, stirring constantly, until the sauerkraut is heated through and onion is softened, 5 to 8 minutes. Taste and season with freshly ground black pepper as needed.

Return the kielbasa to the skillet and toss to combine. Garnish with fresh thyme leaves or finely chopped parsley leaves if using.