Preheat oven to 375 degrees. Grease cookie sheets.
In a bowl, using a mixer, combine butter, sugar, vanilla, salt and flour. Mix well. Dough will be crumbly, but moist.
Place a heaping teaspoonful of dough in palm of your hand. Flatten a little, than place cherry half in center of dough. Form dough around cherry and roll dough into a ball.
Place on prepared cookie sheet. Bake in preheated oven for 15 minutes. Do not let cookies brown. They should be light in color. Cool slightly on wire cooling rack. While baking, blend confectioners' sugar and butter in a bowl, with a mixer, on low speed. Gradually add milk, a teaspoon at a time, until a frosting that spreads easily forms. Add vanilla and beat until creamy.
While cookies are still very warm, frost tops of cookies and place a piece of cherry on top of frosting Makes 4 dozen cookies. Enjoy!