Heat a large skillet. Add the oil. When it begins to smoke lightly, add the onions. Season them with salt and pepper. Add the butter and some of the water and cook, stirring occasionally, until the onions are tender, 10-15 minutes.
Finish and serve:
Add the brown sugar and the thyme. Stir to blend. Cook for an additional 5 minutes. Add the vinegar and reduce until almost all of the liquid has evaporated. Taste for seasoning.