Preheat your oven to 350°F (180°C). Grease and line an 8x8-inch baking pan with parchment paper.

In a medium bowl, whisk together the flour, baking soda, salt, ginger, cinnamon, and cloves.

In a separate bowl or in a liquid measuring cup, whisk the molasses and hot water together.

In a large bowl, using a handheld mixer or a stand mixer fitted with the paddle attachment, beat the butter on high speed until smooth and creamy, about 1 minute. Add the brown sugar and continue beating on high for another 2 minutes, until the mixture is well combined and slightly fluffy. Scrape down the sides and bottom of the bowl as needed.

Add the egg and vanilla and mix on medium-high speed until fully incorporated, pausing again to scrape the bowl. With the mixer on low, add the dry ingredients in three additions, alternating with the hot water–molasses mixture. Begin and end with the dry ingredients, mixing just until each addition is incorporated—do not overmix. Give the batter a quick whisk by hand to ensure there are no large lumps (a few small ones are fine). The batter will be slightly thick.

Pour the batter into the prepared pan and bake for 30–40 minutes, or until the cake is fully set. Check for doneness by inserting a toothpick into the center—if it comes out clean with just a few moist (not wet) crumbs, the cake is ready. Transfer to a wire rack and cool for at least 30 minutes.

Make the frosting: In a large bowl using a handheld mixer or stand mixer fitted with a whisk or paddle attachment, beat the cream cheese and butter together on medium-high speed until creamy and no lumps remain, about 2 minutes. Add confectioners’ sugar, vanilla extract, and pinches of cinnamon. Beat on low speed for 20 seconds, then increase to high speed until everything is completely combined and the frosting is creamy. Spread on top of the cooled cake.

Slice and serve.

Cover leftover cake tightly and store in the refrigerator for up to 5 days.