1. Heat the olive oil and butter in a medium heavy-bottomed nonreactive saucepan over medium heat. When it is hot, add the onion and cook for about 15 minutes, stirring as needed, until it is caramelized.
2. Add the apple cider, vinegar, sugar, rosemary sprig, salt, and pepper. Reduce the heat to medium-low and cook for another 30 to 35 minutes, stirring occasionally, until the onion has reached a jam-like consistency.
3. Discard the rosemary sprig and set the jam aside to cool before using, or cover and refrigerate for up to 1 week.
Yield: 1 cup