Cut each grapefruit in half, then trim a thin slice off the bottom so it sits level. With a paring knife or grapefruit knife, cut around each segment; remove any seeds. Place the grapefruit halves on the prepared baking sheet. Combine brown sugar and cinnamon (or cardamom) in a small bowl. Brush each grapefruit half with butter, then sprinkle with about 2 teaspoons of the sugar mixture.