Mix the sugar and cream in heavy medium saucepan. Place over medium high heat. Bring to a boil, stirring constantly so sugar dissolves. Simmer for 6 minutes from the time the mixture boils. Mixture will be frothy.
Remove the pan from the heat and immediately add unsweetened chocolate and the butter. Stir until butter and chocolate melt and mixture is uniform. Stir in the vanilla extract.
Pour liquid frosting into a microwave-safe container or bowl, cover, and refrigerate the frosting until cool but not completely hardened. (If you over-chill, then microwave very briefly so it can be scooped out of the bowl or container.) Process the mixture in a food processor or beat with heavy duty mixer until the frosting is lighter in color and somewhat fluffy. Spread onto cooled cake layers. Makes enough frosting for one 8 - 9 inch two-layer cake.
Notes
This recipe was inspired by the Satin Fudge Frosting recipe in Abby Mandel's book "More Taste than Time" published in 1988 by Simon & Schuster.