Preheat oven to 350 degrees F (175 degrees C).

In a large bowl, cream together the butter and sugar until smooth. Gradually stir in the flour until well blended. On a lightly floured surface, roll the dough out to 1/4 inch thickness. Cut into rounds using a cookie cutter.

Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely. Sandwich one teaspoon of jam between two cookies. Repeat with remaining cookies. In a medium bowl, gradually stir the milk into the confectioners' sugar until the icing is of a spreadable consistency. Spread on top of the sandwiched cookies. Top each cookie with half of a cherry while the icing is still wet.