In a large bowl, cream butter and sugar until light and fluffy. Beat in the eggs, molasses and milk. Combine the flour, cinnamon, baking soda, salt and ginger; gradually add to creamed mixture and mix well (dough will be soft). Cover and refrigerate for at least 1 hour.
Drop by rounded tablespoonfuls 2 in. apart onto greased baking sheets. Bake at 375° for 8-10 minutes or until set. Remove to wire racks to cool.
In a small bowl, beat the filling ingredients until smooth. Spread over the bottoms of half of the cookies; top with remaining cookies. Store in the refrigerator.