Put the milk in a saucepan and whisk in the flour. Bring to a simmer over medium heat and simmer until thickened, about 5 minutes, whisking often. Strain and cool to room temperature.
Cream the butter, sugar, and vanilla in a mixer fitted with a paddle attachment until light and very fluffy. Add the cooled thickened milk and mix until smooth. Drizzle in the melted chocolate and continue whipping until well blended.