Preheat oven to 350°F. Grease and flour 2 (9-inch) round cake pans. Mix flour, baking powder, ginger, baking soda and cinnamon in medium bowl; set aside.
Beat butter, brown sugar and molasses in large bowl with electric mixer on medium speed until well mixed. Add eggs and vanilla; beat well. Add flour mixture alternately with boiling water, beating on low speed after each addition until smooth. Pour into prepared pans.
Bake 18 to 20 minutes or until toothpick inserted in center comes out clean. Cool in pans 10 minutes. Remove from pans; cool completely on wire rack.
For the Frosting, pour milk into medium bowl. Add pudding mix, ginger and cinnamon. Beat with wire whisk 2 minutes. Gently stir in whipped topping. Fill and frost cooled cake with with Frosting. Refrigerate cake until ready to serve.