Preheat the oven to 350°F and prepare a square baking dish with grease and parchment. Set aside.
In a medium-sized bowl, mix together the dry ingredients. Whisk together the flour, baking soda, ginger, cinnamon, cloves, and salt. Set aside.
In a small bowl, combine the hot water with the molasses. Stir the two together until they are fully combined.
In a large bowl with a hand mixer or in a bowl of a stand mixer with a paddle attachment, cream the butter, about 1 minute. Add the brown sugar, making sure to scrape down the sides as needed. Once the butter and brown sugar are combined, add the egg and vanilla.
With the wet ingredients mixed, add in the dry mixture and molasses mixture, alternating back and forth. Start with the dry mixture and make sure to end with it as well. Add in the flour mixture slowly so it won't go all over the place. Try not to over mix the batter, it will be thick and a little lumpy but that is okay.
Pour the batter into the pan and bake the cake for about 35-45 minutes. The cake will be a deep color when baked. Use a toothpick to see if it is done. If it comes out clean, let it cool on a wire rack before frosting.
While the cake is cooling, make the frosting. In a medium-sized bowl or in the stand mixer with the whisk attachment, whisk together the cream cheese and butter until smooth. The mixture should be smooth when ready. Then, add in the confectioner's sugar, (a little at a time otherwise it will be messy!) milk, and vanilla. Continue to mix the mixture until it looks smooth. If you think the frosting is a bit too sweet, add in a pinch of salt to balance out the flavors.
The cake should be eaten fresh but should last about 3-5 in an airtight container and stored in the refrigerator.