Place butter in a microwave-safe mug. Heat in the microwave until melted, 10 to 20 seconds. Whisk in egg, brown sugar, white sugar, molasses, and vanilla extract.

Mix whole wheat flour, wheat germ, ginger, cinnamon, baking soda, and salt together in a small bowl. Add to the mug; stir well until combined.

Cook in the microwave in 40-second intervals until the top springs back when lightly pressed, about 1 minute 20 seconds.

Whisk confectioners' sugar and water together in a small bowl to make icing. Invert cake onto a serving plate and drizzle icing on top.