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Cut the onions root to tip: Slice 1/2-inch off the stem ends of the onions and the roots off of the root end. Place the onions cut side down on the cutting board. Cut them in half through the root end. Peel back the peels from the onions. Lay the onions cut side down and make angled cuts into each onion, aimed at the center, cutting almost all the way, but not completely through the root end. Make the cuts to your desired level of thickness.The root end will help hold the onion together as you cut it, making it easier to cut. Then cut a V in the root end to cut out the tough root holding the slices together.
Heat olive oil and butter, add onion slices: Use a wide, thick-bottomed sauté pan for maximum pan contact with the onions. Coat the bottom of the pan with olive oil, or a mixture of olive oil and butter (about 1 teaspoon per onion). Heat the pan on medium high heat until the oil is shimmering. Add the onion slices and stir to coat the onions with the oil. Spread the onions out evenly over the pan and let cook, stirring occasionally. Depending on how strong your stovetop burner is you may need to reduce the heat to medium or medium low to prevent the onions from burning or drying out.After 10 minutes, sprinkle some salt over the onions, and if you want, you can add some sugar to help with the caramelization process. (I add only about a teaspoon of sugar for 5 onions, you can add more.) One trick to keeping the onions from drying out as they cook is to add a little water to the pan.
Cook, stirring every few minutes: Let cook for 30 minutes to an hour more, stirring every few minutes. As soon as the onions start sticking to the pan, let them stick a little and brown, but then stir them before they burn. The trick is to leave them alone enough to brown (if you stir them too often, they won't brown), but not so long so that they burn. After the first 20 to 30 minutes you may want to lower the stove temperature a little, and add a little more oil, if you find the onions are verging on burning.A metal spatula will help you scrape up the browned bits from the bottom of the pan as the caramelization proceeds. As the onions cook down, you may find you need to scrape the pan every minute, instead of every few minutes.Continue to cook and scrape, cook and scrape, until the onions are a rich, browned color. At the end of the cooking process you might want to add a little balsamic vinegar or wine to help deglaze the pan and bring some additional flavor to the onions.Store refrigerated for several days in an air-tight container.