In a 12" skillet over medium-high, heat the oil until it immediately bubbles when an onion touches it (~ a minute)
Add the onions, sprinkle with salt, and cook, stirring frequently, until they've wilted completely and are beginning to stick to the bottom of the pan (~ 10 -20 minutes)
Much of the onion's moisture will evaporate and the pan will begin to brown. The cooking time will vary depending upon the moisture content of your onions.
Reduce the heat to medium-low and continue to cook, stirring and scraping the pan with a wooden spoon every few minutes. (If the pan begins to look like it’s burning, add a couple of tablespoons of the liquid, stir, and lower the heat a bit.)
Continue the process of cooking, stirring, scraping and adding water as needed, until the onions have turned a uniform caramel brown (~ another 35-40 minutes)
Add a couple of tablespoons of the liquid and scrape the pan well.
If you're not using the onions immediately, let them cool in the pan, then transfer them to a storage container.
Caramelized onions can be kept refrigerated for around a week or frozen for up to three months.