In a large bowl using a whisk, beat butter, sugar, and molasses just until smooth. Add grated ginger, orange zest, rum, and egg and beat until combined.
In a medium bowl, whisk flours, baking soda, salt, and spices until combined. Gradually fold dry ingredients into wet ingredients, until dough just comes together.
Preheat oven to 350° and line 2 large baking sheets with parchment paper. Using a tablespoon and dampened hands, portion dough into 1” balls and place 2” apart on prepared sheets.
Bake until puffed and set, 12 to 14 minutes.
Meanwhile, make glaze: In a medium bowl, whisk powdered sugar, cardamom, and cream until smooth, adding more cream to loosen, if necessary.
Let cookies cool for 10 minutes then dip the tops of each cookie in the glaze. Let glaze harden before eating.
