In the skillet, warm the olive oil over medium heat and add the onions. Cook until caramelized around the edges, about 20 minutes.
While the onions cook, bring water to boil in the large pot. Add a bit of salt and the green beans. Cook uncovered until crisp-tender, about 6 to 7 minutes. Drain.
Add the broth to the onions and stir, cooking for about 5 more minutes. Add the balsamic vinegar, brown sugar, salt, and pepper, and stir.
Add the green beans to the skillet and cover. Cook for about 2 minutes. Serve warm.