Brush cut side of figs with butter and sprinkle with sugar.
Preheat stovetop grill.
Brush grill surface with butter.
Grill figs, cut side down, over medium heat, until golden brown (about 1 1/2 minutes).
Watch that sugar doesn't burn.
Grill only until sugar melts and figs are warm.
Remove figs from grill and place cut side up around chicken livers.