Position an oven rack about 4 inches below the broiler and heat the broiler on high. With a serrated knife, cut the grapefruit in even halves. Using a small paring knife or a grapefruit knife, cut each section away from the surrounding membrane. Set the grapefruit halves in a shallow broiler-safe pan (such as an enameled baking dish or a heavy-duty rimmed baking sheet). If necessary, trim a thin slice off their bottoms so they sit level.

In a small bowl, stir together the honey and the vanilla extract. Drizzle the honey mixture over the grapefruit halves. Dust each with a bit of cardamom. Broil until bubbling and lightly browned in spots, 4 to 6 minutes. Remove from the oven and let cool slightly. Serve warm or at room temperature.