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 browned and caramelized, 12 to 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 minutes before serving. The dip will keep, refrigerated, overnight.
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