Preheat the oven to 350 degrees F and line a baking sheet with parchment paper. In a medium bowl, whisk together the flour, ground ginger, cinnamon, baking soda, and salt. In a separate bowl, beat the butter and brown sugar until creamy. Add the molasses, egg, and vanilla extract, then mix until smooth. Gradually stir the dry ingredients into the wet mixture until a dough forms. Chill the dough for 30 minutes. Roll the dough out on a floured surface to about 1/4 inch thick and cut into 8 rectangles. Place the rectangles on the prepared baking sheet and bake for 10 to 12 minutes until the edges firm up. Let the cookies cool completely on a wire rack. In a small bowl, stir the powdered sugar and milk into a smooth icing, then add green food coloring until the shade looks bright. Spread the green icing over each cookie and press two candy eyes into the icing while it is still wet.