flour and milk based frosting

Thanks for all this additional info, Hilarybeee

This kind of frosting is sometimes called either Ermine frosting or German Buttercream. I have a cooked milk frosting on my bakery's menu. I like to use a combination of heavy cream and whole milk, but you can use just whole milk. I love the smooth, velvety texture. I especially love that it is more stable at room temperature than whipped cream. However, it can be very tricky. If stability becomes an issue, try a recipe that utilizes cornstarch. Overbeating will cause the frosting to become soup. If this happens, put the mixer bowl in the fridge for an hour or two. Then re-beat. The frosting will hold up at room temp, but I prefer to keep cakes like this in the refrigerator until about an hour before they will be served.
There is a good recipe in Baked Explorations for Grass Hopper Bars with an ermine frosting here:
http://www.seriouseats.com/recipes/2010/12/grasshopper-bars-brownies-holiday-baking-recipe.html

I think that Stella Parks' recipe for Gilt Taste is a particularly good albeit modified German Buttercream that incorporates cream cheese and custard, making it more stable.
http://www.gilttaste.com/stories/2292-the-original-red-wine-velvet-cake-recipe

Thanks so much for this response--exactly what I was looking to know. Now I'm just waiting for an excuse to test it out.

This is an old recipe that makes a delectable frosting with a very light texture--
almost like whipped cream. It is very different from a buttercream in its lightness, yet it has a rich taste. Get past the weirdness of incorporating cooked flour and milk into a frosting and you will be delighted with the result.
The key is to beat it until it is extremely smooth , and if you see any lumps just keep beating. Try this on a cake where you normally would use cream cheese frosting to see what you think.