Preheat oven to 400°. In a large bowl, cream shortening and sugar until light and fluffy. Beat in eggs and vanilla. Combine flour, baking powder and salt; gradually add to creamed mixture and mix well.
Shape dough into 1-1/2-in. balls. Place 1 in. apart on ungreased baking sheets. Bake 8-10 minutes or until lightly browned. Remove to wire racks to cool.
For icing, in a small bowl, combine milk, butter, vanilla and confectioners' sugar until smooth. Tint with food coloring if desired. Dip tops of cookies in icing; allow excess to drip off. Sprinkle with coarse sugar if desired. Place on waxed paper; let stand until set.