Does this type of frosting have a name?
Looks like the term you want is "French buttercream". 

American buttercream is, as you say, full of sugar but rarely has eggs, relying on powdered sugar and butter with a bit of wet in the form of a dash of milk, cream, or alcohol.

French buttercream is based on egg yolks and contains relatively little sugar and is cooked over a double boiler.


  French buttercream is made by heating a sugar syrup until it reaches soft ball stage and then whipping it into beaten egg yolks and soft butter. This is the shiny, rich frosting of your dreams.


Italian Buttercream is similar but uses egg whites rather than yolks, so it's more of a meringue.


  It's similar to French buttercream in that you use a boiling syrup of sugar and water and cook it to soft-ball stage but then you pour it over beaten egg whites to create a meringue. As the meringue cools, you slowly add butter and mix like your life depended on it.