In a large bowl, whisk together the coconut milk, lime zest, lime juice, minced garlic, grated ginger, brown sugar, salt, and black pepper to create a marinade.
Place the chicken breasts into the marinade, ensuring they are fully coated. Cover and refrigerate for at least 1 hour (or overnight for more flavor).
Preheat your grill or skillet over medium-high heat and add the olive oil.
Remove the chicken from the marinade, allowing the excess to drip off, and place on the grill or skillet. Cook for 6-7 minutes on each side, or until the chicken reaches an internal temperature of 165°F (74°C) and has nice grill marks.
Once cooked, remove the chicken from the heat and let it rest for 5 minutes. This allows the juices to redistribute.
Slice the chicken and serve it warm, drizzling some reserved marinade that was cooked to a simmer (note: only reserved what hasn't touched raw chicken) over the top.
Garnish with freshly chopped cilantro for a burst of color and flavor.