Heat the olive oil in a small skillet over medium heat. Add the shallots and 1/2 teaspoon of salt and cook, stirring frequently, until translucent and soft, about 5 minutes. Add 1/4 cup of water and the balsamic vinegar, reduce the heat to medium and cook, stirring occasionally, until the shallots are caramelized, about 15 minutes.
Combine the shallots, yogurt, milk, chives, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl and stir until blended. Cover and chill for 30 at least minutes before serving.