Peel and slice onions to desired thickness.

Heat a wide, heavy-bottomed pan, and add the olive oil and butter (if using) until shimmering.

Add the onion slices and stir to coat the onions with the oil. Spread the onions out evenly over the pan and let cook, stirring occasionally.

Turn heat to medium low. Stir the onions every few minutes to prevent sticking to the pan. Let the onions cook for 30 minutes to 1 hour more, stirring every few minutes. The onions will sweat and release their own liquid.

As soon as the onions start brown, stir them before they burn. Add a little more oil as needed, if you find the onions are sticking or starting to burn.

Use a wooden or metal spatula to scrape up the browned bits from the bottom of the pan as the caramelization proceeds. Continue to cook and scrape until the onions are a rich, brown color.

At the end of the cooking process add a splash of balsamic vinegar or wine to deglaze the pan, and continue to cook on low until all the added liquid has been absorbed, and onions are a thick jammy texture.