tasty donuts with vanilla filling

All you need is:

1 package regular or quick-acting yeast

1/8 cup water 

3/4 cup lukewarm milk (scalded then cooled)

1/4 cup granulated sugar

1/2 teaspoon salt

1 egg

1/6 cup shortening

2 1/2 cups all-purpose flour

 Filling:

1/4 cup solid vegetable shortening

1/4 cup butter or margarine

1/2 teaspoon clear vanilla extract

2 cups sifted confectioners' sugar

1 tablespoon milk