Place a medium pan over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add the onion, thyme, some salt and pepper to the pan, and put a lid on it. Cook the onions, stirring occasionally, until softened, about 10 minutes. Remove the lid and continue cooking the onions until deep golden brown, about 20 minutes (if the onions begin to stick to the pan too much, add a splash of water and scrape up any bits that are sticking). Transfer the onions to a bowl and cool.