Start cooking the onions. Set a large, heavy-bottomed pan or cast-iron skillet over medium heat. Add the olive oil and butter and swirl to coat the bottom of your pan. Add the sliced onions, mixing to coat the onions in butter and oil.
Continue cooking the onions. Allow your onions to cook, stirring occasionally. If the onions start to brown, immediately reduce heat to medium-low heat. After about 10 minutes, sprinkle with a pinch of salt and stir to combine. By this point, your onions should be softening and lightly golden.
Continue to cook, stirring often. Continue to cook for an additional 30-45 minutes, stirring every 2-3 minutes to prevent them from burning. Avoid stirring continuously. Stir just enough to allow the onions to cook and stick to the bottom of the skillet, browning just slightly – but not burning. If you stir too often, they won’t brown. If you don’t stir often enough, they will burn
Use a metal spatula to scrape the pan often. After about 30 minutes, your onions should be a rich golden color. Use a metal spatula to scrape up the brown bits from the bottom of your skillet. Continue to cook, stir, and scrape the brown bits (fond) from the bottom of the pan until your onions are a dark rich golden brown color.