Preheat grill (or a grill pan) to medium heat. Mix brown sugar and cinnamon in a small bowl. Set aside.
Lightly brush the cut side of peaches with melted butter.
Place peaches, cut side down, onto grill pan. Grill for about 3 minutes, or until peaches start to soften just a little and grill marks form.
Flip peaches over and sprinkle tops with cinnamon brown sugar mixture.
Grill for an additional 2-3 minutes, or until sugar mixture begins to caramelize. Remove peaches from grill pan.