Slice onions in thin rings, melt butter into oil and swirl together. Add onions and cook over low heat until browned and translucent (about 15 minutes). Add vinegar and brown sugar and stir until well-mixed. Boil the mixture on medium-high heat and then slow simmer, uncovered and stirring occasionally, for about 10 minutes, or until the liquid has evaporated and the onion has a deep color.