Bring broth to a boil in a large saucepan. Stir in grits, and return to a boil. Cover, reduce heat, and simmer 5 minutes, or until grits are thickened, stirring occasionally. Stir in 1/2 cup cheese and salt. Remove from heat. Spoon grits into a greased 9 by 13-inch baking pan. Cover, and chill at least 2 hours, or until firm. Un-mold grits onto a large cutting board.
Preheat oven to 400 degrees F.
Cut out 48 (1 1/2-inch) circles using a round or fluted cookie cutter. Brush a large jellyroll pan with melted butter. Place grits rounds on pan. Bake for 15 minutes. Turn grits, and bake 15 minutes more. Set aside. (Up to this point, the recipe can be prepared ahead. If preparing early, cover and refrigerate grit rounds until you are ready to top with mushroom mixture.)
In a bowl, stir together the mayonnaise, 1 cup Parmesan, mushrooms, and onion rings. Top each grits round evenly with mushroom mixture. Broil 5 minutes, or until lightly browned and heated through.