In a bowl, whisk together balsamic vinegar, olive oil, brown sugar, and garlic. Add chicken to the bowl and toss to combine. Let marinate for 20 minutes.
In a blender, add the sour cream, cilantro, milk, and lime juice. Blend until smooth. Season with salt to taste. Refrigerate until ready to use.
Preheat a grill or grill pan on medium heat. Grill the chicken, intermittently brushing with the marinade, until cooked through to an internal temperature of 160 degrees using a thermometer, (time depends on the thickness of the breast).
Drizzle the sauce over the chicken and serve. Garnish with cilantro. Serve with bread.