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Beat butter in large bowl with electric mixer on medium speed until light and fluffy. Add vanilla and melted chocolate; mix well.
Stir cinnamon into confectioners' sugar. Gradually add to butter mixture, beating until well blended after each addition and scraping sides and bottom of bowl frequently.
Beat in milk until light and fluffy. If frosting is too thick to spread, gradually beat in additional milk. Store in refrigerator up to 2 weeks. Rewhip before using.