Peel the onions, rinse, and dry them. Cut them in half through the root. Remove the root, position the onions with the root end facing you, and slice them into ¼-inch slices.
Heat the butter over medium heat in a large, deep skillet. Tilt the skillet to spread the butter over the bottom.
When the butter starts foaming, add the onions in two or three batches, sprinkling salt on each batch and allowing it to slightly soften before adding the next batch.
Lower the heat to medium-low. Cook the onions, uncovered, stirring them often, until they are very soft and deeply browned. This should take about 45 minutes. If the pan becomes too hot and there are brown bits on its bottom, add a splash of water and use it to deglaze the pan. You can keep adding water as needed or lower the heat to low.
Remove the onions to a plate. Serve immediately.
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Notes
You can make this recipe with red onions. In fact, all types of onions can be successfully caramelized - yellow, white, red, and also sweet onions such as Vidalia.
Although oil works in this recipe, the milk solids in butter help with the browning process, similar to the role they play when blackening food.
There's no need to add sugar. Onions contain natural sugars that promote the browning process.
You can keep the leftovers in an airtight container in the fridge for up to five days. Reheat them in the microwave, covered, at 50% power. You can also freeze the leftovers for up to three months. I like to freeze them in single-serve portions.