Heat oil in a large skillet over medium heat until hot. Add onions and sprinkle with salt. Cook and stir for 15 minutes or until moisture from onions has evaporated and onions are completely wilted.
Reduce heat to medium-low. Sprinkle vinegar over onions. Cook and stir for 20 minutes or until lightly golden. Stir in brown sugar; cook and stir for 15-20 minutes longer or until onions are a caramel brown color.
If onions begin to stick to skillet, add water, 1 tablespoon at a time, until onions no longer stick to skillet.