Cream butter. Add powdered sugar, beating until creamy. Blend the egg in well, then blend in the flour gradually. Set aside for 30 minutes. Do no chill.
Divide dough into 2 or 3 parts, depending on the size of your breadboard. Place one at a time on the lightly floured board. Roll out to about ¼ inch in thickness. Cut with a floured round cookie cutter and place the cookies on very, very lightly floured cookie sheets.
Bake 10 minutes in a preheated 375° oven, or until cookies are browned very lightly on the edges.
Remove from cookie sheets and place about ¾ teaspoon of the filling on each of ½ the total number of cookies. Top each filled cookie with a plain one.
Scald ¾ cup of milk in top of a double boiler until a thin film appears.
Sift flour, sugar and salt into a saucepan. Add to this the remaining ¼ cup of milk. Then add the scalded milk gradually, stirring constantly. Wash the rim of the double boiler to remove any scum of milk there may be left. Bring mixture rapidly to a boil, stirring constantly, and let boil for 3 minutes.
Transfer mixture back to the top of the double boiler and let cook over boiling water 5 minutes more.
Stir 3 tablespoons of this hot mixture into the beaten egg yolks. When the milk and flour combination has cooled somewhat, blend the egg yolks all at once into it and cook over low heat 3 to 5 minutes until smooth, stirring constantly.
Remove from flame and stir in the butter and the vanilla. Cover and set aside to cool, stirring now and then.
If you wish, you may frost the filled cookies as follows: