Preheat oven to 350 degrees F. To make brown butter, melt butter in a small skillet over medium heat. When the butter is completely melted, whisk or stir fairly consistently. It will start to crackle and then foam. Immediately remove the skillet from the heat, and pour into a mixing bowl when the butter begins to foam, give off a nutty aroma and turn a light caramel brown color. Keep your eye on it - this can take anywhere from 4-8 minutes.
Add the sugar to the brown butter and use an electric stand or hand-held mixer to mix on low speed to start, then increase the speed and beat for one minute. (Mixture will NOT be light and fluffy.) Add in egg, and beat until combined. In a separate small bowl, stir together mashed bananas with baking soda and let sit for 2 minutes. Add mashed banana and vanilla to the sugar mixture and beat until combined.
In a separate bowl, combine flour, salt, cinnamon, nutmeg and cloves. Add the dry ingredients to the wet ingredients gradually, in three or so batches, mixing just until combined before adding the next batch.
Drop small spoonfuls of batter onto a baking sheet that is lined with parchment paper or a silicone mat and bake for about 10 minutes, or until cookies are golden brown around the edges. Let cool on baking sheet for 5 minutes, then move to a cooling rack to cool completely.
For the icing: Follow the directions for browning the butter in step 1. Once butter is browned, remove from heat and whisk in powdered sugar, and drizzle icing over cooled cookies. You can then add a sprinkle of finely chopped pecans or walnuts.