1. Preheat oven to 350°F (175°C). Line an 8×8-inch pan with parchment paper and grease the sides.

2. Whisk together flour, ginger, cinnamon, cloves, baking soda, and salt in a medium bowl.

3. Beat softened butter and brown sugar until creamy. Add eggs one at a time, mixing well after each addition.

4. Stir in molasses, sourdough starter, and milk until the mixture is glossy.

5. Fold the dry ingredients into the wet mixture in two additions, alternating with hot water, mixing until just combined.

6. Pour batter into the prepared pan and bake for 35 to 40 minutes or until a toothpick inserted in the center comes out clean.

7. Cool the cake in the pan for 10 minutes, then transfer to a wire rack to cool completely.

8. Beat cream cheese and softened butter until smooth. Gradually add powdered sugar and vanilla, mixing until creamy.

9. Spread the icing evenly over the cooled cake and chill for 15 minutes before slicing for clean edges.

10. Slice into squares and serve chilled or at room temperature.