Preheat oven to 350°.
Coat a 9-inch round cake pan with cooking spray; line bottom of pan with wax paper. Coat wax paper with cooking spray; set aside.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, ginger, cinnamon, baking soda, and salt in a bowl, stirring with a whisk.
Place brown sugar and butter in a large bowl, and beat with a mixer at medium-high speed until well blended (about 3 minutes). Add the egg, and beat well. Beat in the applesauce and molasses (batter may look slightly curdled). Reduce mixing speed to low. Add one-third of the flour mixture, and beat just until blended. Repeat procedure with remaining flour mixture. Add Guinness stout, and beat just until combined.
Scrape batter into prepared pan. Bake at 350° for 30 minutes or until a toothpick inserted in the center comes out clean. Cool in pan on a wire rack for 10 minutes; remove from pan. Cool completely. Spread cake with whipped topping. Cut into 8 wedges.