In a pan over medium-high heat, add olive oil, sliced onions, salt, and half a cup of water. Saute until water is evaporated, then turn heat down to low
Saute onions until they start to turn caramel in color, then add sugar. Allow sugar to melt before stirring.
Stir continuously until sugar reduces down by about half and thickens, about 10 minutes. Test it on the cold plate to see if it sets. Once it is the texture you desire in your jam, it is ready. Turn off the heat, and stir in balsamic vinegar.
Allow to cool, then store in sterilized mason jars.