Bring water to boil in a saucepan large enough to hold all the onions. When the water starts boiling, add the onions, bring back to a boil and cook until tender, about 20 minutes. Use a small paring knife to pierce them and make sure they're soft all the way through. Drain the onions in a colander.
Heat the butter in a large heavy skillet over medium-high heat. Add the onions and stir to coat them with butter. Season with salt and cook, stirring frequently, until caramelized, about 5 minutes.
Add the vinegar and sprinkle the sugar over the onions. Shake the skillet back and forth to make the onions roll around a bit. Bring the vinegar to a boil and cook another 5 minutes, until the syrup thickens slightly.
Serve warm or at room temperature with the syrup drizzled over the onions.