In a small saucepan, cook balsamic vinegar and sugar over medium heat until thickened and reduced to 3 tablespoons, about 6 minutes.
Meanwhile, in a large skillet, bring 5cm water to a boil over medium heat. Add vinegar and salt. Crack eggs one at a time and cook for about 3 minutes.
Divide the asparagus between 4 bowls, drizzle with the balsamic vinegar and sugar mixture, add the eggs, parmesan, and serve.