While the patties are cooking, preheat a pan over medium heat and add olive oil. Then, add sliced onions to the hot pan and cook them until they start to soften. After a few minutes, add balsamic vinegar, unsalted butter, brown sugar, and salt. Continue to cook the onions, stirring occasionally, until they shrink and caramelize, which takes about 15-20 minutes. For fully caramelized onions, let them cook for at least 30 minutes.