Cook cornmeal in salted water according to package directions, then stir in rosemary. Meanwhile, grease a large shallow baking dish with butter. Pour in hot polenta and spread evenly. Let cool. When cool, cut polenta into diamonds, fingers, circles or other shape.
Preheat grill to 400°F. Brush the polenta pieces with melted butter, then cook over grids until they are slightly brown with marks from the grill. Sprinkle with parmesan cheese and serve immediately.