Cut the tops and bottoms off the onions and slice each one in half lengthwise, down through the root end. Peel off the papery skin and any tough outer layers, and cut out any hard root left at the bottom with a small V-shaped cut.

Set each half cut side down with the root end facing you and slice it into strips about ¼ inch thick, along the lines that run from the root to the tip. Cutting with the grain helps the slices soften without falling apart over the long cook. When you're a little over halfway through, flip the smaller piece onto its side so it's easier to hold.

Melt the butter in a large, heavy pot over medium heat. Add enough onions to fill the pot about two-thirds of the way, add the salt and give it a good stir.

As the onions give up their liquid and cook down, keep adding more until they're all in the pot. Stir about once every 10 minutes; the liquid keeps the onions on the bottom from burning, but they can still scorch if you leave them too long.

Keep cooking on medium until the onions have given up all their liquid, usually 60 to 90 minutes and sometimes longer. To check, drag a spoon across the bottom of the pot: if liquid seeps into the gap, keep going.

Once the liquid is gone, turn the heat down to low, or somewhere between low and medium-low. Stir about once every 5 minutes, each time scraping up the brown bits (the fond) from the bottom of the pot and stirring them back into the onions.

Keep a close eye on the pot toward the end, when the brown bits can burn very quickly; if any scorch, the onions will taste bitter. The whole process usually takes 4 to 5 hours.

The onions are done when they're an even dark brown, the color of a Tootsie Roll. Pour off any melted butter pooled in the pot.