Add the olive oil to a medium-sized nonstick skillet, over medium heat.
Once the oil is hot, add the sliced onions. Use a spatula to stir for a minute and coat them lightly in the oil.
Then slowly add enough water to cover the bottom the pan, up about a 1/4" - add the cover and let the onions steam for 5 minutes.
After 5 minutes, check again and stir. Let the water cook down. Once it has evaporated, add more water, cover and cook again for 5 minutes. You will repeat this process until the onions have caramelized and reduced down. They should reduce by over half. They will turn a darker brown color and become very soft and sweet. Keep sweating them with the water and cooking down until they are sweet enough to your liking.
Set aside until ready to assemble your sandwiches. The caramelized onions store well in the fridge for a couple days.