In a medium bowl, whisk together balsamic vinegar, olive oil, brown sugar, garlic, thyme, rosemary, and season with salt and pepper. Reserve ¼ of the mixture on the side.
Add chicken to the bowl and toss to combine. Let marinate at least 20 minutes or up to overnight.
In a blender, add the sour cream, cilantro, 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 reserved marinade, until cooked through to an internal temperature of 160 degrees using a thermometer, about 5 minutes per side (time depends on the thickness of the breast).
Drizzle the sour cream mixture over the chicken and serve.