This frosting uses flour to thicken it and keep it together, rather than the cooked protein structure of egg whites (Classic Butter Cream and 7 Minute Frosting), or the strong bond between butter and air (Classic Butter Cream and Quick Vanilla Butter Cream). After making all four frosting recipes, we've done most types except a Swiss Buttercream which is similar to a 7 minute, where you heat the egg whites with sugar, but for not as long and then you add creamed butter to it. There are also ganache frostings which start with melted chocolate and blend liquids into it like cream, corn syrup, vanilla, etc., and are never whipped, to make a shiny more dense chocolate frosting. There's also cream cheese frostings where you beat the cream cheese until very fluffy and add ingredients for sweetness and flavor.
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.