Beat egg with a fork in a small, heavy saucepan. Mix in sugar and milk. Heat on medium-low heat, whisking constantly, until it just starts to boil. Continue beating rapidly until mixture thickens to a thin custard consistency. Remove from heat and cool completely, about 1 hour.

Beat butter in a bowl with an electric mixer until light and creamy, about 5 minutes. Add cooled custard, 1 to 2 tablespoons at a time, beating well after each addition. Beat in rum and vanilla extract. Keep beating until buttercream is very light and fluffy.