Heat oil in a medium skillet over medium. Add shallots and season generously with salt and pepper. Cook, stirring occasionally and reducing heat if needed, until shallots are golden brown and tender, 15–18 minutes.
Mix shallots, vinegar, and yogurt in a medium bowl; season with salt and pepper.
Stir 1 tablespoon of chives into yogurt and top with more chives. Serve with chips, crackers, or toast points for dipping.