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 Parmesan 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. Cut out 48 (1 1/2-inch) circles using a round or fluted cookie cutter (Paula used a clean tomato paste can as a cookie cutter).
Brush a large jellyroll pan with melted butter. Place grit 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 shrimp mixture.).
In a large bowl, combine cream cheese and half-and-half, stirring until combined. Stir in cheese, parsley, and shrimp. Top each grits round evenly with shrimp mixture. Top mixture with grated Cheddar. Broil 5 minutes, or until lightly browned and heated through.