Beat butter and sugar for cake in a bowl with an electric mixer until creamy. Add egg and mix well. Mix in molasses.
Sift flour, ginger, baking soda, cinnamon, cloves, and salt together in another bowl. Stir into butter mixture. Add coffee and beat well.
Grease and flour the inside of a small, quart-sized French baking dish or baking insert. Pour in batter. Place in the slow cooker. Pour water around the insert or baking dish, halfway up the sides. Cover insert with lid, or cover baking dish with paper towels.
Cover cooker with lid slightly ajar to allow excess moisture to escape. Cook until edges are golden and a knife inserted into the center comes out clean, on Low, 3 to 4 hours, or High, 1 3/4 to 2 hours. Remove dish from the cooke and let cool on a wire rack for at least 5 minutes before running a knife around the outer edge of the cake and inverting onto a serving plate.
While cake cools, combine powdered sugar, water, lemon juice, cornstarch, and salt for lemon sauce in a saucepan, stirring with each addition. Cook over medium heat until thick and bubbly, about 1 minute. Remove from the heat and stir in butter.