Uncover and continue to cook, stirring occasionally at first and then more frequently once the onions start to brown, until the onions are a rich caramel brown color, 35 to 45 minutes. Reduce the heat as needed if the bottom of the pot is getting too dark. Make sure to scrape the bottom of the pot when stirring. Add 1/2 cup dry white wine and cook, stirring and scraping the bottom of the pot, until reduced by about half, about 30 seconds. Remove and discard the thyme stems.