Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking dish.
In a large mixing bowl, whisk together the flour, brown sugar, ginger, cinnamon, nutmeg, baking soda, and salt until well combined.
In another bowl, mix the molasses, milk, melted butter, eggs, and vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and mix until just combined.
Gradually stir in the hot water until the batter is smooth. The batter will be thin.
Pour the batter into the prepared baking dish.
Bake for 30-35 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool for about 10 minutes before serving.