Preheat oven to 350 degrees F. and line a 8 x 8 inch baking dish with parchment paper.
Place melted butter and brown sugar into a large mixing bowl. Stir until well combined. Add molasses and hot water, stirring until combined. Beat egg in a small bowl then add to wet ingredients, stirring to combine. Add flour, cinnamon, ginger, salt and baking soda, stirring until combined. Transfer cake batter to baking dish and bake for 30-32 minutes, until cake is baked through. Use a toothpick to check center of cake. Remove from oven and let cool completely.
To prepare icing, place powdered sugar into a large bowl. Slowly stir in heavy cream until icing is creamy and smooth for drizzling. Add more heavy cream as needed to thin icing. Drizzle icing over cake and cut into squares to serve.