Preheat the oven to 350°F / 175°C. Butter and line an 8" square pan* with parchment paper.
In a small saucepan set over medium heat, brown the butter. Stir the butter occasionally until it turns amber in color and develops little brown specks.
Transfer the butter to a small bowl and set it in the freezer for 5 minutes to chill.
Add the banana and gingerbread spice to a medium sized mixing bowl. Mash the banana to your desired consistency.
Add in the cooled brown butter, brown sugar, molasses, both eggs, the vanilla, and the sour cream. Whisk vigorously until the mixture is well combined.
Add in the flour, baking powder, and salt. Mix together until only a few flour streaks remain. Add in the chopped walnuts, then mix just until no flour streaks remain.
Pour the cake batter into the prepared cake pan. Bake for 30-35 minutes, or until a toothpick inserted in the center of the cake comes out clean.