In a large mixing bowl, cream butter and sugar. Add egg; mix well. Beat in molasses. Combine flour, baking soda, salt, cinnamon and ginger; add to creamed mixture and mix well. (The dough will be very stiff.) Cover and chill several hours or overnight.
On a lightly floured surface, roll dough to 1/8 inch thickness; cut into 2-1/4 inch to 2-1/2 inch circles. Place 1/2 tsp. preserves on half of the circles; top with remaining circles. Pinch edges together to seal. Place on greased baking sheets.
Bake at 350 degrees F for 10 minutes or until lightly browned. Cool on wire rack. Combine glaze ingredients and frost cooled cookies.