Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan.
In a medium bowl, whisk together the flour, ginger, cinnamon, nutmeg, baking soda, and salt. Set aside.
In a large bowl, cream the softened butter and brown sugar together until light and fluffy, about 3-4 minutes.
Beat in the egg, molasses, and vanilla extract until well combined.
Gradually add the dry ingredients to the wet mixture, mixing until just combined. Do not overmix.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 25-30 minutes, or until a toothpick inserted in the center comes out clean.
Allow the gingerbread to cool in the pan for about 10 minutes, then transfer to a wire rack to cool completely.
While the gingerbread cools, prepare the molasses frosting. In a medium bowl, beat the softened butter until creamy.
Gradually add the powdered sugar, molasses, vanilla extract, and salt, mixing until smooth and fluffy.
Once the gingerbread is completely cool, spread the molasses frosting evenly over the top.