Preheat oven to 375F.
Arrange chicken breasts on an oven-proof baking dish or a baking pan. Loosen the skin from the meat with your fingers such that you have 2 sides attached. Slice the goat cheese into 8 thick pieces. Place 2 basil leaves, then two pieces of goat cheese under the skin of each chicken breast. Rub each piece with olive oil and sprinkle with kosher salt and freshly ground black pepper.
Bake the breasts for about 35-45 minutes, until the internal temperature reaches 160F. You may want to finish by turning on the broiler for a couple of minutes to better brown and crisp up the skin. I usually turn on the broiler (on high) once the internal temperature of the chicken reaches 150F. Make sure to watch the chicken closely as the skin will brown fairly quickly. If that happens before the chicken is done, turn off the broiler and move the baking dish a couple of positions down.
Remove from the oven and let rest for 5 minutes to allow the meat re-absorb some of the lost moisture before serving.