Mix mayonnaise, brown sugar, and cinnamon in a small bowl. Lightly baste the cut side of each peach half with the mayo mixture. Prepare grill; clean grates thoroughly before placing fruit on the grill. Grill peaches cut-side down until fruit has grill marks and begins to soften, about 2 to 3 minutes. Remove from grill and top with feta cheese and sliced basil. Serve warm.