Preheat oven to 350°F and grease a 9x9-inch baking pan.
In a bowl, whisk together the flour, ginger, cinnamon, cloves, salt, and baking soda.
In a separate large bowl, mix the melted butter, brown sugar, and molasses until smooth.
Stir in the hot water, then add the egg and vanilla extract.
Gradually mix in the dry ingredients until well combined.
Pour the batter into the prepared pan and bake for 35-40 minutes, or until a toothpick inserted in the center comes out clean.
Let the cake cool completely before frosting.
To make the frosting, beat the cream cheese and butter until creamy.
Add the powdered sugar, lemon juice, lemon zest, and vanilla extract. Beat until smooth.
Spread the frosting over the cooled cake and serve.