1. Cut the top and bottom off each onion and peel the onions. Cut vertically though the middle (not so you have rings); place the cut side down on a cutting board and slice into a julienne. Repeat with all the onions.
2. Place the onions into a saucepan over low heat. Sprinkle with a little salt, cover and cook, stirring occasionally, allowing the onions to sweat out their liquid until they become very soft and translucent, being careful not to caramelize or burn them; about 20 minutes. Remove the cover and add half (1/4 cup) of the balsamic; continue to cook, uncovered, until the liquid coats the back of a spoon, about 15 to 20 minutes more.
3. Meanwhile, in a separate pan over medium high heat, pour the sugar into the center of the pan and surround with water. Stir the sugar into the water to moisten it; cook, shaking the pan occasionally, until the moisture is expelled, keeping a close eye on the mixture as it changes color and begins to caramelize. When the sugar is light amber, remove from the heat and stir in the remaining balsamic until well incorporated. Add the caramel sauce to the cooked onions, taste and adjust seasonings with salt. Cool and serve.