Pour the olive oil into a Dutch oven or casserole over a medium heat.
Chop the onions into large pieces. Depending on the size of the onions, I usually cut them in eighths. Add them to the hot olive oil.
After a few minutes, reduce the heat. Add the minced garlic and then let simmer another minute.
Add the thyme, salt and pepper. Let the onions simmer covered until very soft and lightly brown. Stir from time to time to prevent the onions from sticking.
Add the white wine and mix so it cleans the bottom of the casserole. Let the alcohol evaporate.
Add the broth and continue to cook over a low heat for about 20 minutes. The broth should amply cover the onions. Add more broth, as needed, until you have the thickness that you like. This recipe will give you a thicker consistency.
Put the grated cheese in a bowl and let everyone add it to their soup. Adding the cheese separately allows each person to take the amount they like. It's also better so that the cheese is not in the casserole when you want to reheat any leftovers the next day.
Serve the soup and enjoy. You can serve with warm bread if you like.