In a saucepan over medium heat, combine the 1/2 cup brown sugar with the water and stir until dissolved. Turn off the heat and add the lime juice and basil leaves. Let stand for at least 15 minutes or up to 1 hour, then discard the basil leaves. Light a grill or preheat a grill pan over medium heat; lightly oil the grates. Lightly sprinkle the peaches with brown sugar and grill cut side down until dark brown grill marks form, about 2 minutes per cut side. Transfer the peaches to bowls and season with a little black pepper. Spoon some of the syrup on top, sprinkle with basil leaves and serve.