Combine the brown sugar, oil, soy sauce, and vinegar in a bowl. Whisk to combine.
Add the marinade to a container that can be sealed airtight or a zip top style bag.
Pound the chicken breasts to even thickness using a meat mallet (or even a rolling pin works).
Place the chicken in the container with the marinade.
Seal up the container and mix around to make sure all of the chicken is covered.
Place in the refrigerator for 2-24 hours.
When you're ready to grill, remove the chicken from the refrigerator and preheat the grill to medium heat (about 375ºF-400ºF).
Once the grill is preheated, oil the grates. Place the chicken breasts on the grill for about 7 minutes per side depending on thickness. Cook until the internal temperature reaches 165°F.
Remove them from the grill and let them rest for a few minutes to allow the juices to redistribute (it makes a big difference!).